I’ve written about intuitionistic logic before. In fact, there’s a whole section about it in my book. But now that I’m reading a lot about type theory, I’m starting to look at it diferently.
When you study classical axiomatic set theory, you’re necessarily also studying classical first order predicate logic. You have to be doing that, because classical axiomatic set theory is deeply and intimately intertwined with FOPL. Similarly, the semantics of FOPL as it’s used in modern math are inextricably tangled with set theory. Sets are specified by predicates; predicates get their meaning from the sets of objects that they satisfy.
You can view type theory – or at least Martin-Loff’s intuitionistic type theory – as having nearly the same relationship to intuitionistic logic. We’ll see that in detail in later posts, but for now, intuitionistic type theory is a fundamental mathematical framework which is built on intuitionistic logic. So you can’t talk about this kind of type theory unless you understand the basics of the logic.
In this post, I’m going to try to explain what intuitionistic logic is, and how it differs from FOPL. (We’ll see all of this in more detail later.)
Intuitionistic logic is a modal predicate logic, which is built around a constructivist idea of truth. The intuitionistic idea of truth ends up being much stronger than what most of us are used to from standard FOPL: it means that nothing exists unless there is a concrete way of constructing it.
For a concrete example of what that means: in standard FOPL with the ZFC axioms, you can prove the Banach-Tarski paradox. Banach-Tarski (which I wrote about HERE) says that it’s possible to take a sphere the size of an orange, cut it into pieces, and then re-assemble those pieces into two spheres the same size as the original orange. Or, alternatively, that you can take those pieces that you sliced an orange-sized sphere into, and re-assemble them into a sphere the size of the sun.
Many people would say that this is, clearly, ridiculous. Others would point out a variety of rationalizations: that a sphere the size of an orange and a sphere the size of the sun contain the same number of points; or that the slicing process transitioned from a metric topology to a collection of non-metric topologies, or several other possible explanations.
But what no one can dispute is that there is one very important property of this proof. Those slices are unconstructable. That is, they exist based on a proof using the axiom of choice, but the sets of points in those topologies can’t be constructed by any process. They exist as a necessary implication of the axiom of choice, but we can’t construct them, and even given a pair of sets, one of which is one of those slices, and one of which isn’t, we can’t identify which one is.
According to intuitionism, this is ridiculous. Saying that something exists, but that it is forever beyond our reach is foolishness. If we can’t construct it, if we can’t describe how to identify it, what does it mean to say that it must exist?
When you’re working in intuitionistic logic, every proof that a type of thing exists consists of either a concrete example of the thing, or a process for constructing an example of the thing. A proof of a negative is a concrete counterexample, or a process for creating one. In computer-sciency-terms, the process doesn’t need to terminate. You don’t have to be able to construct something in finite time. But you need to have a process that describes how to contsruct it. So you can, for example, still do Cantor’s diagonalization in intuitionistic logic: if someone gives you an alleged complete 1:1 mapping between the real numbers and the integers, the proof tells you how to create a counterexample. But you can’t do the proof of Banach-Tarski, because it relies on an axiom-of-choice existence proof of something non-constructable.
The way that intuitionistic logic creates that constructivist requirement is not what you might expect. When I first heard about it, I assumed that it was based on a statement of principle: a proof has to create a concrete example. But that approach has an obvious problem: how do you mathematically define it? Logic is supposed to be purely symbolic. How can you take an abstract statement about what a proof should be, and make it work in logic?
Logic is built on inference rules. You have a collection of statements, and a collection of rules about how to use those statements to produce proofs. It turns out that by making a couple of simple changes to the rules of inference that you can get exactly the constructivist requirements that we’d want. It’s based on two real changes compared to standard FOPL.
Intuitionistic logic is modal. In FOPL, any given statement is either true or false. If it’s not true, then it’s false. If it’s true, it’s always true, and always was true. There’s no other choice. In intuitionistic logic, that’s not really the case: intuitionistic logic has three states: true, false, and unknown. If you know nothing about it, then it’s formally unknown, and it will stay unknown until there’s a proof about it; once you find a proof, it’s truth value changes from unknown to either true or false. All of the inference rules of intuitionistic logic only allow inference from proven statements. You can’t reason about an unknown – you need to have a proof that moves it from unknown to either true or false first.
The semantics of this are quite simple: it’s a tiny change in the definition of truth. In FOPL, a statement is true if there exists a proof of that statement, and it’s false if there’s a proof of the negation of that statement. In intuitionistic logic, a statement is true if you have a proof of that statement; and it’s false if you can prove that there is no proof of the statement If you haven’t proven , then is unknown. If is unknown, then is also unknown. is, similarly, not true until you have a proof of either or : it means that either “There is a proof of A or there is a proof of “. But if we don’t know if there’s a proof of either one, then it’s unknown! You could argue that this is true in FOPL as well – but in FOPL, you can rely on the fact that , and you can use that in a proof, and explore both options. In intuitionistic logic, you can’t: you can’t do anything with until you’ve got a proof.
It’s amazing how small the change to FOPL is to produce something that is so strongly constructionist. The easiest way to appreciate it is to just look at the rules, and how they change. To do that, I’m going to quickly walk through the inference rules of intuitionistic logic, and then show you what you’d need to change to get classical FOPL. Most of the time, when I’ve written about logics, I used sequents to write the inference rules; for ease of typesetting (and for the fun of doing something just a bit different), this time, I’m going to use Hilbert calculus (the same method that Gödel used in his incompleteness proof.) In HC, you define axioms and inference rules. For intuitionistic logic, we need to define three inference rules:
Modus Ponens: Given and , you can infer .
Universal Generation: Given , you can infer if is not free in .
Existential Generation: Given , you can infer , if is not free in .
With the inference rules out of the way, there’s a collection of axioms. Each axiom is actually a schema: you can substitute any valid statement for any of the variables in the axioms.
Then-1: .
Then-2:
And-1:
And-2:
And-3:
Or-1:
Or-2:
Or-3:
False: . (For a bit of explanation, this rule means that we don’t need to have rules – can be treated as .)
Universal: , if is not bound by instantiating .
Existential: if is not bound by instantiating .
That’s intuitionistic logic. What’s the difference between that and FOPL? What kinds of powerful reasoning features did you need to give up from FOPL to get this strongly constructivist logic?
Just one simple axiom: the law of the excluded middle, .
That’s it. Get rid of the excluded middle, and you’ve got the beautiful constructivist intuitionistic logic. All we had to give up is one of the most intuitionnally obvious rules in all of logic.
After a bit of a technical delay, it’s time to finish the repost of incompleteness! Finally, we’re at the end of our walkthrough of Gödel great incompleteness proof. As a refresher, the basic proof sketch is:
Take a simple logic. We’ve been using a variant of the Principia Mathematica’s logic, because that’s what GÃ¶del used.
Show that any statement in the logic can be encoded as a number using an arithmetic process based on the syntax of the logic. The process of encoding statements numerically is called GÃ¶del numbering.
Show that you can express meta-mathematical properties of logical statements in terms of arithemetic properties of their GÃ¶del numbers. In particular, we need to build up the logical infrastructure that we need to talk about whether or not a statement is provable.
Using meta-mathematical properties, show how you can create an unprovable statement encoded as a GÃ¶del number.
What came before:
Gödel numbering: The logic of the Principia, and how to encode it as numbers. This was step 1 in the sketch.
Arithmetic Properties: what it means to say that a property can be expressed arithemetically. This set the groundwork for step 2 in the proof sketch.
Encoding meta-math arithmetically: how to take meta-mathematical properties of logical statements, and define them as arithmetic properties of the Gödel numberings of the statements. This was step 2 proper.
So now we can move on to step three, where we actually see why mathematical logic is necessarily incomplete.
What I did in the last post was walk through a very laborious process that showed how we could express meta-mathematical properties of logical statements as primitive recursive functions and relations. Using that, we were able to express a non-primitive-recursive predicate provable, which is true for a particular number if and only if that number is the Gödel number representation of a statement which is provable.
pred provable(x) =
some y {
proofFor(y, x)
}
}
The reason for going through all of that was that we really needed to show how we could capture all of the necessary properties of logical statements in terms of arithmetic properties of their Gödel numbers.
Now we can get to the target of Gödel’s effort. What Gödel was trying to do was show how to defeat the careful stratification of the Principia’s logic. In the principia, Russell and Whitehead had tried to avoid problems with self-reference by creating a very strict type-theoretic stratification, where each variable or predicate had a numeric level, and could only reason about objects from lower levels. So if natural numbers were the primitive objects in the domain being reasoned about, then level-1 objects would be things like specific natural numbers, and level-1 predicates could reason about specific natural numbers, but not about sets of natural numbers or predicates over the natural numbers. Level-2 objects would be sets of natural numbers, and level-2 predicates could reason about natural numbers and sets of natural numbers, but not about predicates over sets of natural numbers, or sets of sets of natural numbers. Level-3 objects would be sets of sets of natural numbers… and so on.
The point of this stratification was to make self-reference impossible. You couldn’t make a statement of the form “This predicate is true”: the predicate would be a level-N predicate, and only a level N+1 predicate could reason about a level-N predicate.
What Gödel did in the arithmetic process we went through in the last post is embed a model of logical statements in the natural numbers. That’s the real trick: the logic of the principia is designed to work with a collection of objects that are a model of the natural numbers. By embedding a model of logical statements in the natural numbers, he made it possible for a level-1 predicate (a predicate about a specific natural number) to reason about any logical statement or object. A level-1 predicate can now reason about a level-7 object! A level-1 predicate can reason about the set defined by a level-1 predicate: a level-1 predicate can reason about itself!. A level-1 predicate can, now, reason about any logical statement at all – itself, a level-2 predicate, or a level-27 predicate. Gödel found a way to break the stratification.
Now, we can finally start getting to the point of all of this: incompleteness! We’re going to use our newfound ability to nest logical statements into numbers to construct an unprovable true statement.
In the last post, one of the meta-mathematical properties that we defined for the Gödel-numbered logic was immConseq, which defines when some statement x is an immediate consequence of a set of statements S. As a reminder, that means that x can be inferred from statements in S in one inferrence step.
We can use that property to define what it means to be a consequence of a set of statements: it’s the closure of immediate consequence. We can define it in pseudo-code as:
def conseq(κ) = {
K = κ + axioms
added_to_k = false
do {
added_to_k = false
for all c in immConseq(K) {
if c not in K {
add c to K
added_to_k = true
}
}
} while added_to_k
return K
}
In other words, Conseq(κ) is the complete set of everything that can possibly be inferred from the statements in κ and the axioms of the system. We can say that there’s a proof for a statement x in κ if and only if x ∈ Conseq(κ).
We can take the idea of Conseq use that to define a strong version of what it means for a logical system with a set of facts to be consistent. A system is ω-consistent if and only if there is not a statement a such that: a ∈ Conseq(κ) ∧ not(forall(v, a)) ∈ Conseq(κ).
In other words, the system is ω-consistent as long as it’s never true that both a universal statement and it. But for our purposes, we can treat it as being pretty much the same thing. (Yes, that’s a bit hand-wavy, but I’m not trying to write an entire book about Gödel here!)
(Gödel’s version of the definition of ω-consistency is harder to read than this, because he’s very explicit about the fact that Conseq is a property of the numbers. I’m willing to fuzz that, because we’ve shown that the statements and the numbers are interchangable.)
Using the definition of ω-consistency, we can finally get to the actual statement of the incompleteness theorem!
Gödel’s First Incompleteness Theorem: For every ω-consistent primitive recursive set κ of formulae, there is a primitive-recursive predicate r(x) such that neither forall(v, r) nor not(forall(v, r)) is provable.
To prove that, we’ll construct the predicate r.
First, we need to define a version of our earlier isProofFigure that’s specific to the set of statements κ:
pred isProofFigureWithKappa(x, kappa) = {
all n in 1 to length(x) {
isAxiom(item(n, x)) or
item(n, x) in kappa or
some p in 0 to n {
some q in 0 to n {
immedConseq(item(n, x), item(p, x), item(q, x))
}
}
} and length(x) > 0
}
This is the same as the earlier definition – just specialized so that it ensures that every statement in the proof figure is either an axiom, or a member of κ.
We can do the same thing to specialize the predicate proofFor and provable:
pred proofForStatementWithKappa(x, y, kappa) = {
isProofFigureWithKappa(x, kappa) and
item(length(x), x) = y
}
pred provableWithKappa(x, kappa) = {
some y {
proofForStatementWithKappa(y, x, kappa)
}
}
If κ is the set of basic truths that we can work with, then provable in κ is equivalent to provable.
Now, we can define a predicate UnprovableInKappa:
pred NotAProofWithKappa(x, y, kappa) = {
not (proofForKappa(x, subst(y, 19, number(y))))
}
Based on everything that we’ve done so far, NotAProofWithKappa is primitive recursive.
This is tricky, but it’s really important. We’re getting very close to the goal, and it’s subtle, so let’s take the time to understand this.
Remember that in a Gödel numbering, each prime number is a variable. So 19 here is just the name of a free variable in y.
Using the Principia’s logic, the fact that variable 19 is free means that the statement is parametric in variable 19. For the moment, it’s an incomplete statement, because it’s got an unbound parameter.
What we’re doing in NotAProofWithKappa is substituting the numeric coding of y for the value of y‘s parameter. When that’s done, y is no longer incomplete: it’s unbound variable has been replaced by a binding.
With that substitution, NotAProofWithKappa(x, y, kappa) is true when xdoes not prove that y(y) is true.
What NotAProofWithKappa does is give us a way to check whether a specific sequence of statements x is not a proof of y.
We want to expand NotAProofWithKappa to something universal. Instead of just saying that a specific sequence of statements x isn’t a proof for y, we want to be able to say that no possible sequence of statements is a proof for y. That’s easy to do in logic: you just wrap the statement in a “∀ x ( )”. In Gödel numbering, we defined a function that does exactly that. So the universal form of provability is: ∀ a (NotAProofWithKappa(a, y, kappa)).
In terms of the Gödel numbering, if we assume that the Gödel number for the variable a is 17, and the variable y is numbered as 19, we’re talking about the statement p = forall(17, ProvableInKappa(17, 19, kappa).
p is the statement that for some logical statement (the value of variable 19, or y in our definition), there is no possible value for variable 17 (a) where a proves y in κ.
All we need to do now is show that we can make p become self-referential. No problem: we can just put number(p) in as the value of y in UnprovableInKappa. If we let q be the numeric value of the statement UnprovableInKappa(a, y), then:
r = subst(q, 19, p)
i = subst(p, 19, r)
i says that there is no possible value x that proves p(p). In other words, p(p) is unprovable: there exists no possible proof that there is no possible proof of p!
This is what we’ve been trying to get at all this time: self-reference! We’ve got a predicate y which is able to express a property of itself. Worse, it’s able to express a negative property of itself!
Now we’re faced with two possible choices. Either i is provable – in which case, κ is inconsistent! Or else i is unprovable – in which case κ is incomplete, because we’ve identified a true statement that can’t be proven!
That’s it: we’ve shown that in the principia’s logic, using nothing but arithmetic, we can create a true statement that cannot be proven. If, somehow, it were to be proven, the entire logic would be inconsistent. So the principia’s logic is incomplete: there are true statements that cannot be proven true.
We can go a bit further: the process that we used to produce this result about the Principia’s logic is actually applicable to other logics. There’s no magic here: if your logic is powerful enough to do Peano arithmetic, you can use the same trick that we demonstrated here, and show that the logic must be either incomplete or inconsistent. (Gödel proved this formally, but we’ll just handwave it.)
Looking at this with modern eyes, it doesn’t seem quite as profound as it did back in Gödel’s day.
When we look at it through the lens of today, what we see is that in the Principia’s logic, proof is a mechanical process: a computation. If every true statement was provable, then you could take any statement S, and write a program to search for a proof of either S or ¬ S, and eventually, that program would find one or the other, and stop.
In short, you’d be able to solve the halting problem. The proof of the halting problem is really an amazingly profound thing: on a very deep level, it’s the same thing as incompleteness, only it’s easier to understand.
But at the time that Gödel was working, Turing hadn’t written his paper about the halting problem. Incompletess was published in 1931; Turing’s halting paper was published in 1936. This was a totally unprecedented idea when it was published. Gödel produced one of the most profound and surprising results in the entire history of mathematics, showing that the efforts of the best mathematicians in the world to produce the perfection of mathematics were completely futile.
On to the next part of Gödel’s proof of incompleteness. To refresh your memory, here’s a sketch of the proof:
Take a simple logic. We’ve been using a variant of the Principia Mathematica’s logic, because that’s what Gödel used.
Show that any statement in the logic can be encoded as a number using an arithmetic process based on the syntax of the logic. The process of encoding statements numerically is called Gödel numbering.
Show that you can express meta-mathematical properties of logical statements in terms of arithemetic properties of their Gödel numbers. In particular, we need to build up the logical infrastructure that we need to talk about whether or not a statement is provable.
Using meta-mathematical properties, show how you can create an unprovable statement encoded as a Gödel number.
What we’ve done so far is the first two steps, and part of the third. In this post, we saw the form of the Principia’s logic that we’re using, and how to numerically encode it as a Gödel numbering. We’ve start started on the third point in this post, by figuring out just what it means to say that things are encoded arithmetically. Now we can get to the part where we see how to encode meta-mathematical properties in terms of arithmetic properties of the Gödel numbering. In this post, we’re going to build up everything we need to express syntactic correctness, logical validity, and provability in terms of arithmetical properties of Gödel numbers. (And, as a reminder, I’ve been using this translation on Gödel’s original paper on incompleteness.)
This is the most complex part of the incompleteness proof. The basic concept of what we’re doing is simple, but the mechanics are very difficult. What we want to do is define a set of predicates about logical statements, but we want those predicates to be expressed as arithmetic properties of the numerical representations of the logical statements.
The point of this is that we’re showing that done in the right way, arithmetic is logic – that doing arithmetic on the Gödel numbers is doing logical inference. So what we need to do is build up a toolkit that shows us how to understand and manipulate logic-as-numbers using arithmetic. As we saw in the last post, primitive recursion is equivalent to arithmetic – so if we can show how all of the properties/predicates that we define are primitive recursive, then they’re arithmetic.
This process involves a lot of steps, each of which is building the platform for the steps that follow it. I struggled quite a bit figuring out how to present these things in a comprehensible way. What I ended up with is writing them out as code in a pseudo-computer language. Before inventing this language, I tried writing actual executable code, first in Python and then in Haskell, but I wasn’t happy with the clarity of either one.
Doing it in an unimplemented language isn’t as big a problem as you might think. Even if this was all executable, you’re not going to be able to actually run any of it on anything real – at least not before you hair turns good and gray. The way that this stuff is put together is not what any sane person would call efficient. But the point isn’t to be efficient: it’s to show that this is possible. This code is really all about searching; if we wanted to be efficient, this could all be done in a different representation, with a different search method that was a lot faster – but that wolud be harder to understand.
So, in the end, I threw together a simple language that’s easy to read. This language, if it were implemented, wouldn’t really even be Turing complete – it’s a primitive recursive language.
Basics
We’ll start off with simple numeric properties that have no obvious connection to the kinds of meta-mathematical statements that we want to talk about, but we’ll use those to define progressively more and more complex and profound properties, until we finally get to our goal.
# divides n x == True if n divides x without remainder.
pred divides(n, x) = x mod n == 0
pred isPrime(0) = False
pred isPrime(1) = False
pred isPrime(2) = True
pred isPrime(n) = {
all i in 2 to n {
not divides(i, n)
}
}
fun fact(0) = 1
fun fact(n) = n * fact(n - 1)
Almost everything we’re going to do here is built on a common idiom. For anything we want to do arithmetically, we’re going to find a bound – a maximum numeric value for it. Then we’re going to iterate over all of the values smaller than that bound, searching for our target.
For example, what’s the nth prime factor of x? Obviously, it’s got to be smaller than x, so we’ll use x as our bound. (A better bound would be the square root of x, but it doesn’t matter. We don’t care about efficiency!) To find the nth prime factor, we’ll iterate over all of the numbers smaller than our bound x, and search for the smallest number which is prime, which divides x, and which is larger than the n-1th prime factor of x. We’ll translate that into pseudo-code:
fun prFactor(0, x) = 0
fun prFactor(n, x) = {
first y in 1 to x {
isPrime(y) and divides(y, x) and prFactor(n - 1, x) < y
}
}
Similarly, for extracting values from strings, we need to be able to ask, in general, what's the nth prime number? This is nearly identical to prFactor above. The only difference is that we need a different bound. Fortunately, we know that the nth prime number can't be larger than the factorial of the previous prime plus 1.
fun nthPrime(0) = 0
fun nthPrime(n) = {
first y in 1 to fact(nthPrime(n - 1)) + 1 {
isPrime(y) and y > nthPrime(n - 1))
}
}
In composing strings of Gödel numbers, we use exponentiation. Given integers x and n, x^{n}, we can obviously compute them via primitive recursion. I'll define them below, but in the rest of this post, I'll write them as an operator in the language:
fun pow(n, 0) = 1
fun pow(n, i) = n * pow(n, i - 1)
String Composition and Decomposition
With those preliminaries out of the way, we can get to the point of defining something that's actually about one of the strings encoded in these Gödel numbers. Given a number n encoding a string, item(n, x) is the value of the nth character of x. (This is slow. This is really slow! We're getting to the limit of what a very powerful computer can do in a reasonable amount of time. But this doesn't matter. The point isn't that this is a good way of doing these things: it's that these things are possible. To give you an idea of just how slow this is, I started off writing the stuff in this post in Haskell. Compiled with GHC, which is a very good compiler, using item to extract the 6th character of an 8 character string took around 10 minutes on a 2.4Ghz laptop. In the stuff that follows, we'll be using this to extract characters from strings that could be hundreds of characters long!)
fun item(n, x) = {
first y in 1 to x {
divides(prFactor(n, x) ** y, y) and
not divides(prFactor(n, x)**(y+1), x)
}
}
Given a string, we want to be able to ask how long it is; and given two strings, we want to be able to concatenate them.
fun length(x) = {
first y in 1 to x {
prFactor(y, x) > 0 and prFactor(y + 1, x) == 0
}
}
fun concat(x, y) = {
val lx = length(x)
val ly = length(y)
first z in 1 to nthprime(lx + ly)**(x + y) {
(all n in 1 to lx {
item(n, z) == item(n, x)
}) and (all n in 1 to ly {
item(n + lx, z) == item(n, y)
})
}
}
fun concatl([]) = 0
fun concatl(xs) = {
concat(head(xs), concatl(tail(xs)))
}
fun seq(x) = 2**x
We want to be able to build statements represented as numbers from other statements represented as numbers. We'll define a set of functions that either compose new strings from other strings, and to check if a particular string is a particular kind of syntactic element.
# x is a variable of type n.
pred vtype(n, x) = {
some z in 17 to x {
isPrime(z) and x == n**z
}
}
# x is a variable
pred isVar(x) = {
some n in 1 to x {
vtype(n, x)
}
}
fun paren(x) =
concatl([gseq(11), x, gseq(13)])
# given the Gödel number for a statement x, find
# the Gödel number for not x.
fun gnot(x) =
concat(gseq(5), paren(x))
# Create the number for x or y.
fun gor(x, y) =
concatl([paren(x), seq(7), paren(y)])
# Create the number for 'forall x(y)'.
fun gforall(x, y) =
concatl([seq(9), seq(x), paren(y)])
# Create the number for x with n invocations of the primitive
# successor function.
fun succn(0, x) = x
fun succn(n, x) = concat(seq(3), succn(n - 1, x))
# Create the number n using successor and 0.
fun gnumber(n) = succn(n, seq(1))
# Check if a statement is type-1.
pred stype_one(x) = {
some m in 1 to x {
m == 1 or (vtype(1, m) and x == succn(n, seq(m))
}
}
# Check if a statement is type n.
pred fstype(1, x) = stype_one(x)
pred fstype(n, x) =
some v in 1 to x {
vtype(n, v) and R(v)
}
}
That last function contains an error: the translation of Gödel that I'm using says R(v) without defining R. Either I'm missing something, or the translator made an error.
Formulae
Using what we've defined so far, we're now ready to start defining formulae in the basic Principia logic. Forumlae are strings, but they're strings with a constrained syntax.
pred elFm(x) = {
some y in 1 to x {
some z in 1 to x {
some n in 1 to x {
stype(n, y) and stype(n+1, z) and x == concat(z, paren(y))
}
}
}
}
All this is doing is expressing the grammar rule in arithmetic form: an elementary formula is a predicate: P(x), where x is a variable on level n, and P is a variable of level x + 1.
The next grammar rule that we encode this way says how we can combine elementary formulae using operators. There are three operators: negation, conjunction, and universal quantification.
pred op(x, y, z) = {
x == gnot(y) or
x == gor(y, z) or
(some v in 1 to x { isVar(v) and x == gforall(v, y) })
}
And now we can start getting complex. We're going to define the idea of a valid sequence of formulae. x is a valid sequence of formulae when it's formed from a collection of formulae, each of which is either an elementary formula, or is produced from the formulae which occured before it in the sequence using either negation, logical-or, or universal quantification.
In terms of a more modern way of talking about it, the syntax of the logic is a grammar. A formula sequence, in this system, is another way of writing the parse-tree of a statement: the sequence is the parse-tree of the last statement in the sequence.
pred fmSeq(x) = {
all p in 0 to length(x) {
elFm(item(n, x)) or
some p in 0 to (n - 1) {
some q in 0 to (n - 1) {
op(item(n,x), item(p, x), item(q, x))
}
}
}
}
The next one bugs me, because it seems wrong, but it isn't really! It's a way of encoding the fact that a formula is the result of a well-defined sequence of formulae. In order to ensure that we're doing primitive recursive formulae, we're always thinking about sequences of formulae, where the later formulae are produced from the earlier ones. The goal of the sequence of formula is to produce the last formula in the sequence. What this predicate is really saying is that a formula is a valid formula if there is some sequence of formulae where this is the last one in the sequence.
Rephrasing that in grammatical terms, a string is a formula if there is valid parse tree for the grammar that produces the string.
pred isFm(x) = {
some n in 1 to nthPrime(length(x)**2)**(x*length(x)**2) {
fmSeq(n)
}
}
So, now, can we say that a statement is valid because it's parsed according to the grammar? Not quite. It's actually a familiar problem for people who write compilers. When you parse a program in some language, the grammar doesn't usually specify variables must be declared before they're used. It's too hard to get that into the grammar. In this logic, we've got almost the same problem: the grammar hasn't restricted us to only use bound variables. So we need to have ways to check whether a variable is bound in a Gödel-encoded formula, and then use that to check the validity of the formula.
# The variable v is bound in formula x at position n.
pred bound(v, n, x) = {
isVar(v) and isFm(x) and
(some a in 1 to x {
some b in 1 to x {
some c in 1 to x {
x == concatl([a, gforall(v, b), c]) and
isFm(b) and
length(a) + 1 ≤ n ≤ length(a) + length(forall(v, b))
}
}
})
}
# The variable v in free in formula x at position n
pred free(v, n, x) = {
isVar(v) and isFm(x) and
(some a in 1 to x {
some b in 1 to x {
some c in 1 to x {
v == item(n, x) and n ≤ length(x) and not bound(v, n, x)
}
}
})
}
pred free(v, x) = {
some n in 1 to length(x) {
free(v, n, x)
}
}
To do logical inference, we need to be able to do things like replace a variable with a specific infered value. We'll define how to do that:
# replace the item at position n in x with y.
fun insert(x, n, y) = {
first z in 1 to nthPrime(length(x) + length(y))**(x+y) {
some u in 1 to x {
some v in 1 to x {
x == concatl([u, seq(item(n, x)), v]) and
z == concatl([u, y, v]) and
n == length(u) + 1
}
}
}
}
There are inference operations and validity checks that we can only do if we know whether a particular variable is free at a particular position.
# freePlace(k, v, k) is the k+1st place in x (counting from the end)
# where v is free.
fun freePlace(0, v, x) = {
first n in 1 to length(x) {
free(v, n, x) and
not some p in n to length(x) {
free(v, p, x)
}
}
}
fun freePlace(k, v, x) = {
first n in 1 to freePlace(n, k - 1, v) {
free(v, n, x) and
not some p in n to freePlace(n, k - 1, v) {
free(v, p, x)
}
}
}
# number of places where v is free in x
fun nFreePlaces(v, x) = {
first n in 1 to length(x) {
freeplace(n, v, x) == 0
}
}
In the original logic, some inference rules are defined in terms of a primitive substitution operator, which we wrote as subst[v/c](a) to mean substitute the value c for the variable c in the statement a. We'll build that up on a couple of steps, using the freePlaces function that we just defined.
# Subst1 replaces a single instance of v with y.
fun subst'(0, x, v, y) = x
fun subst1(0k, x, v, y) =
insert(subst1(k, x, v, y), freePlace(k, v, x), y)
# subst replaces all instances of v with y
fun subst(x, v, y) = subst'(nFreePlaces(v, x), x, v, y)
The next thing we're going to do isn't, strictly speaking, absolutely necessary. Some of the harder stuff we want to do will be easier to write using things like implication, which aren't built in primitive of the Principia logic. To write those as clearly as possible, we'll define the full suite of usual logical operators in terms of the primitives.
# implication
fun gimp(x, y) = gor(gnot(x), y)
# logical and
fun gand(x, y) = gnot(gor(gnot(x), gnot(y)))
# if/f
fun gequiv(x, y) = gand(gimp(x, y), gimp(y, x))
# existential quantification
fun gexists(v, y) = not(gforall(v, not(y)))
Axioms
The Peano axioms are valid logical statements, so they have Gödel numbers in this system. We could compute their value, but why bother? We know that they exist, so we'll just give them names, and define a predicate to check if a value matches them.
The form of the Peano axioms used in incompleteness are:
Zero: ¬(succ(x_{1}) = 0)
Uniqueness: succ(x_{1}) = succ(y_{1}) Rightarrow x = y
const pa1 = ...
const pa2 = ...
const pa3 = ...
pred peanoAxiom(x) =
(x == pa1) or (x == pa2) or (x == pa3)
Similarly, we know that the propositional axioms must have numbers. The propositional
axioms are:
I'll show the translation of the first - the rest follow the same pattern.
# Check if x is a statement that is a form of propositional
# axiom 1: y or y => y
pred prop1Axiom(x) =
some y in 1 to x {
isFm(x) and x == imp(or(y, y), y)
}
}
pred prop2Axiom(x) = ...
pred prop3Axiom(x) = ...
pred prop4Axiom(x) = ...
pred propAxiom(x) = prop2Axiom(x) or prop2Axiom(x) or
prop3Axiom(x) or prop4Axiom(x)
Similarly, all of the other axioms are written out in the same way, and we add a predicate isAxiom to check if something is an axiom. Next is quantifier axioms, which are complicated, so I'll only write out one of them - the other follows the same basic scheme.
The two quantifier axioms are:
quantifier_axiom1_condition(z, y, v) = {
not some n in 1 to length(y) {
some m in 1 to length(z) {
some w in 1 to z {
w == item(m, z) and bound(w, n, y) and free(v, n, y)
}
}
}
}
pred quantifier1Axiom(x) = {
some v in 1 to x {
some y in 1 to x {
some z in 1 to x {
some n in 1 to x {
vtype(n, v) and stype(n, z) and
isFm(y) and
quantifier_axiom1_condition(z, y, v) and
x = gimp(gforall(v, y), subst(y, v, z))
}
}
}
}
}
quanitifier_axiom2 = ...
isQuantifierAxiom = quantifier1Axiom(x) or quantifier2Axiom(x)
We need to define a predicate for the reducibility axiom (basically, the Principia's version of the ZFC axiom of comprehension). The reducibility axiom is a schema: for any predicate , . In our primitive recursive system, we can check if something is an instance of the reducibility axiom schema with:
pred reduAxiom(x) =
some u in 1 to x {
some v in 1 to x {
some y in 1 to x {
some n in 1 to x {
vtype(n, v) and
vtype(n+1, u) and
not free(u, y) and
isFm(y) and
x = gexists(u, gforall(v, gequiv(concat(seq(u), paren(seq(v))), y)))
}
}
}
}
}
Now, the set axiom. In the logic we're using, this is the axiom that defines set equality. It's written as . Set equality is defined for all types of sets, so we need to have one version of axiom for each level. We do that using type-lifting: we say that the axiom is true for type-1 sets, and that any type-lift of the level-1 set axiom is also a version of the set axiom.
fun typeLift(n, x) = {
first y in 1 to x**(x**n) {
all k in 1 to length(x) {
item(k, x) ≤ 13 and item(k, y) == item(k, v) or
item(k, x) > 13 and item(k, y) = item(k, x) * prFactor(1, item(k, x))**n
}
}
}
We haven't defined the type-1 set axiom. But we just saw the axiom above, and it's obviously a simple logical statement. That mean that it's got a Gödel number. Instead of computing it, we'll just say that that number is called sa1. Now we can define a predicate to check if something is a set axiom:
val sa1 = ...
pred setAxiom(x) =
some n in 1 to x {
x = typeLift(n, sa)
}
}
We've now defined all of the axioms of the logic, so we can now create a general predicate to see if a statement fits into any of the axiom categories:
pred isAxiom(x) =
peanoAxiom(x) or propAxiom(x) or quantifierAxom(x) or
reduAxiom(x) or setAxiom(x)
Proofs and Provability!
With all of the axioms expressible in primitive recursive terms, we can start on what it means for something to be provable. First, we'll define what it means for some statement x to be an immediate consequence of some statements y and z. (Back when we talked about the Principia's logic, we said that x is an immediate consequence of y and z if either: y is the formula z ⇒ x, or if c is the formula ∀v.x).
pred immConseq(x, y, z) = {
y = imp(z, x) or
some v in 1 to x {
isVar(v) and x = forall(v, y)
}
}
Now, we can use our definition of an immediate consequence to specify when a sequence of formula is a proof figure. A proof figure is a sequence of statements where each statement in it is either an axiom, or an immediate consequence of two of the statements that preceeded it.
pred isProofFigure(x) = {
(all n in 0 to length(x) {
isAxiom(item(n, x)) or
some p in 0 to n {
some q in 0 to n {
immConseq(item(n, x), item(p, x), item(q, x))
}
}
}) and
length(x) > 0
}
We can say that x is a proof of y if x is proof figure, and the last statement in x is y.
pred proofFor(x, y) =
isProofFigure(x) and
item(length(x), x) == y
Finally, we can get to the most important thing! We can define what it means for something to be provable! It's provable if there's a proof for it!
pre provable(x) =
some y {
proofFor(y, x)
}
}
Note that this last one is not primitive recursive! There's no way that we can create a bound for this: a proof can be any length.
At last, we're done with these definition. What we've done here is really amazing: now, every logical statement can be encoded as a number. Every proof in the logic can be encoded as a sequence of numbers: if something is provable in the Principia logic, we can encode that proof as a string of numbers, and check the proof for correctness using nothing but (a whole heck of a lot of) arithmetic!
Next post, we'll finally get to the most important part of what Gödel did. We've been able to define what it means for a statement to be provable - we'll use that to show that there's a way of creating a number encoding the statement that something is not provable. And we'll show how that means that there is a true statement in the Principia's logic which isn't provable using the Principia's logic, which means that the logic isn't complete.
In fact, the proof that we'll do shows a bit more than that. It doesn't just show that the Principia's logic is incomplete. It shows that any consistent formal system like the Principia, any system which is powerful enough to encode Peano arithmetic, must be incomplete.
In the last post, we saw how to take statements written in the logic of the Principia Mathematica, and convert them into numerical form using Gödel numbering. For the next step in Gödel’s proof, we need to go meta-mathematical.
Ultimately, we want to write first-order statements that can reason about first order statements. But the entire structure of the principia and its logic is designed to make
that impossible. First order statements can only reason about numbers and their properties.
But now, we’ve got the ability to represent statements – first order, second order, third order, any order. What we still need is a way of describing the properties of those numerical statements in terms of operations that can be expressed using nothing but first order statements.
The basic trick to incompleteness is that we’re going to use the numerical encoding of statements to say that a predicate or relation is represented by a number. Then we’re going to write predicates about predicates by defining predicates on the numerical representations of the first-order predicates. That’s going to let us create a true statement in the logic that can’t be proven with the logic.
To do that, we need to figure out how to take our statements and relations represented as numbers, and express properties of those statements and relations in terms of arithmetic. To do that, we need to define just what it means to express something arithmetically. Gödel did that by defining “arithmetically” in terms of a concept called primitive recursion.
I learned about primitive recursion when I studied computational complexity. Nowadays, it’s seen as part of theoretical computer science. The idea, as we express it in modern terms, is that there are many different classes of computable functions. Primitive recursion is one of the basic complexity classes. You don’t need a Turing machine to compute primitive recursive functions – they’re a simpler class.
The easiest way to understand primitive recursion is that it’s what you get in a programming language with integer arithmetic, and simple for-loops. The only way you can iterate is by repeating things a bounded number of times. Primitive recursion has a lot of interesting properties: the two key ones for our purposes here are: number theoretic proofs are primitive recursive, and every computation of a primitive recursive function is guaranteed to complete within a bounded amount of time.
The formal definition of primitive recursion, the way that Gödel wrote it, is quite a bit more complex than that. But it means the same thing.
We start with what it means to define a formula via primitive recursion. (Note the language that I used there: I’m not explaining what it means for a function to be primitive recursive; I’m explaining what it means to be defined via primitive recursion.) And I’m defining formulae, not functions. In Gödel’s proof, we’re always focused on numerical reasoning, so we’re not going to talk about programs or algorithms, we’re going to about the definition of formulae.
A formula is defined via primitive recursion if, for some other formulae and :
Base:
Recursive: .
So, basically, the first parameter is a bound on the number of times that can invoked recursively. When it’s 0, you can’t invoke any more.
A formula is primitive recursive if it defined from a collection of formulae where any formula is defined via primitive recursion from , or the primitive succ function from Peano arithmetic.
For any formula in that sequence, the degree of the formula is the number of other primitive recursive formulae used in its definition.
Now, we can define a primitive recursive property: is primitive recursive if and only if there exists a primitive recursive function such that .
With primitive recursive formulae and relations defined, there’s a bunch of theorems about how you can compose primitive recursive formulae and relations:
Every function or relation that you get by substituting a primitive recursive function for a variable in a primitive recursive function/relation is primitive recursive.
If R and S are primitive relations, then ¬R, R∧S, R∨S are all primitive recursive.
If and are primitive recursive functions, then the relation is also primitive recursive.
Let and be finite-length tuples of variables. If the function and the relation are primitive recursive, then so are the relations:
Let and be finite-length tuples of variables. And let be the smallest value of for which and is true, or 0 if there is no such value. Then if the function and the relation are primitive recursive, then so is the function .
By these definitions, addition, subtraction, multiplication, and integer division are all primitive recursive.
Ok. So, now we’ve got all of that out of the way. It’s painful, but it’s important. What we’ve done is come up with a good formal description of what it means for something to be an arithmetic property: if we can write it as a primitive recursive relation or formula, it’s arithmetic.
The first step in Gödel’s incompleteness proof was finding a way of taking logical statements and encoding them numerically. Looking at this today, it seems sort-of obvious. I mean, I’m writing this stuff down in a text file – that text file is a stream of numbers, and it’s trivial to convert that stream of numbers into a single number. But when Gödel was doing it, it wasn’t so obvious. So he created a really clever mechanism for numerical encoding. The advantage of Gödel’s encoding is that it makes it much easier to express properties of the encoded statements arithmetically. (Arithmetically means something very specific here; we’ll see what in a later post.
Before we can look at how Gödel encoded his logic into numbers, we need to look at the logic that he used. Gödel worked with the specific logic variant used by the Principia Mathematica. The Principia logic is minimal and a bit cryptic, but it was built for a specific purpose: to have a minimal syntax, and a complete but minimal set of axioms.
The whole idea of the Principia logic is to be purely syntactic. The logic is expected to have a valid model, but you shouldn’t need to know anything about the model to use the logic. Russell and Whitehead were deliberately building a pure logic where you didn’t need to know what anything meant to use it. I’d really like to use Gödel’s exact syntax – I think it’s an interestingly different way of writing logic – but I’m working from a translation, and the translator updated the syntax. I’m afraid that switching between the older Gödel syntax, and the more modern syntax from the translation would just lead to errors and confusion. So I’m going to stick with the translation’s modernization of the syntax.
The basic building blocks of the logic are variables. Already this is a bit different from what you’re probably used to in a logic. When we think of logic, we usually consider predicates to be a fundamental thing. In this logic, they’re not. A predicate is just a shorthand for a set, and a set is represented by a variable.
Variables are stratified. Again, it helps to remember where Russell and Whitehead were coming from when they were writing the Principia. One of their basic motivations was avoiding self-referential statements like Russell’s paradox. In order to prevent that, they thought that they could create a stratified logic, where on each level, you could only reason about objects from the level below. A first-order predicate would be a second-level object could only reason about first level objects. A second-order predicate would be a third-level object which could reason about second-level objects. No predicate could ever reason about itself or anything on its on level. This leveling property is a fundamental property built into their logic. The way the levels work is:
Type one variables, which range over simple atomic values, like specific single natural numbers. Type-1 variables are written as , .
Type two variables, which range over sets of atomic values, like sets of natural numbers. A predicate, like IsOdd, about specific natural numbers would be represented as a type-2 variable. Type-2 variables are written , , …
Type three variables range over sets of sets of atomic values. The mappings of a function could be represented as type-3 variables: in set theoretic terms, a function is set of ordered pairs. Ordered pairs, in turn, can be represented as sets of sets – for example, the ordered pair (1, 4) would be represented by the set { {1}, {1, 4} }. A function, in turn, would be represented by a type-4 variable – a set of ordered pairs, which is a set of sets of sets of values.
Using variables, we can form simple atomic expressions, which in Gödel’s terminology are called signs. As with variables, the signs are divided into stratified levels:
Type-1 signs are variables, and successor expressions. Successor expressions are just Peano numbers written with “succ”: 0, succ(0), succ(succ(0)), succ(a_{1}), etc.
Signs of any type greater than 1 are just variables of that type/level.
Once you have signs, you can assemble the basic signs into formulae. Gödel explained how to build formulae in a classic logicians form, which I think is hard to follow, so I’ve converted it into a grammar:
elementary_formula → sign_{n+1}(sign_{n})
formula → ¬(elementary_formula)
formula → (elementary_formula) or (elementary_formula)
formula → ∀ sign_{n} (elementary_formula)
That’s the entire logic! It’s tiny, but it’s enough. Everything else from predicate logic can be defined in terms of combinations of these basic formulae. For example, you can define logical “and” in terms of negation and logical “or”: (a ∧ b) ⇔ ¬ (¬ a ∨ ¬ b).
With the syntax of the system set, the next thing we need is the basic axioms of logical inference in the system. In terms of logic the way I think of it, these axioms include both “true” axioms, and the inference rules defining how the logic works. There are five families of axioms.
First, there’s the Peano axioms, which define the natural numbers.
: 0 is a natural number, and it’s not the successor of anything.
: Successors are unique.
: induction works on the natural numbers.
Next, we’ve got a set of basic inference rules about simple propositions. These are defined as axiom schemata, which can be instantiated for any set of formalae , , and .
Axioms that define inference over quantification. is a variable, is any formula, is any formula where is not a free variable, and is a sign of the same level as , and which doesn’t have any free variables that would be bound if it were inserted as a replacement for .
: if formula is true for all values of , then you can substitute any specific value for in , and must still be true.
The Principia’s version of the set theory axiom of comprehension:
And last but not least, an axiom defining set equivalence:
So, now, finally, we can get to the numbering. This is quite clever. We’re going to use the simplest encoding: for every possible string of symbols in the logic, we’re going to define a representation as a number. So in this representation, we are not going to get the property that every natural number is a valid formula: lots of natural numbers won’t be. They’ll be strings of nonsense symbols. (If we wanted to, we could make every number be a valid formula, by using a parse-tree based numbering, but it’s much easier to just let the numbers be strings of symbols, and then define a predicate over the numbers to identify the ones that are valid formulae.)
We start off by assigning numbers to the constant symbols:
Symbols
Numeric Representation
0
1
succ
3
¬
5
∨
7
∀
9
(
11
)
13
Variables will be represented by powers of prime numbers, for prime numbers greater that 13. For a prime number p, p will represent a type one variable, p^{2} will represent a type two variable, p^{3} will represent a type-3 variable, etc.
Using those symbol encodings, we can take a formula written as symbols x_{1}x_{2}x_{3}…x_{n}, and encode it numerically as the product 2^{x1}3^{x2}5^{x2}…p_{n}^{xn}, where p_{n} is the nth prime number.
For example, suppose I wanted to encode the formula: ∀ x_{1} (y_{2}(x_{1})) ∨ x_{2}(x_{1}).
First, I’d need to encode each symbol:
“∀” would be 9.
“x_{1}“” = 17
“(” = 11
“y_{2}” = 19^{2} = 361
“(” = 11
“x_{1}” = 17
“)” = 13
“∨” = 7
“x_{2}” = 17^{2} = 289
“(” = 11
“x_{1}” = 17
“)” = 13
“)” = 13
The formula would thus be turned into the sequence: [9, 17, 11, 361, 11, 17, 13, 7, 289, 11, 17, 13, 13]. That sequence would then get turned into a single number 2^{9} 3^{17} 5^{11} 7^{361} 11^{11} 13^{17} 17^{13} 19^{7} 23^{289} 29^{11} 31^{17} 37^{13} 41^{13}, which according to Hugs is the number (warning: you need to scroll to see it. a lot!):
Next, we’re going to look at how you can express interesting mathematical properties in terms of numbers. Gödel used a property called primitive recursion as an example, so we’ll walk through a definition of primitive recursion, and show how Gödel expressed primitive recursion numerically.
I’m going to be on vacation this week, which means that I won’t have time to write new posts. But my friend Dr. SkySkull was just talking about Gödel on twitter, and chatting with him, I realized that this would be a good time to repost some stuff that I wrote about Gödel’s incompleteness proof.
Incompleteness is one of the most beautiful and profound proofs that I’ve ever seen. If you’re at all interested in mathematics, it’s something that’s worth taking the effort to understand. But it’s also pretty on-topic for what I’ve been writing about. The original incompleteness proof is written for a dialect of math based on ST type theory!
It takes a fair bit of effort to work through the incompleteness proof, so it’ll be a weeks worth of reposts. What I’m going to do is work with this translation of the original paper where Gödel published his first incompleteness proof. Before we can get to the actual proof, we need to learn a bit about the particular kind of logic that he used in his proof.
It goes right back to the roots of type theory. Set theory was on the rocks, due to Russell’s paradox. Russell’s paradox did was show that there was a foundational problem in math. You could develop what appeared to be a valid mathematicial structure and theory, only to later discover that all the work you did was garbage, because there was some non-obvious fundamental inconsistency in how you defined it. But the way that foundations were treated simple wasn’t strong or formal enough to be able to detect, right up front, whether you’d hard-wired an inconsistency into your theory. So foundations had to change, to prevent another incident like the disaster of early set theory.
In the middle of this, along came two mathematicians, Russell and Whitehead, who wanted to solve the problem once and for all. They created an amazing piece of work called the Principia Mathematica. The principia was supposed to be an ideal, perfect mathematical foundation. It was designed to have to key properties: it was supposed to consistent, and it was supposed to be complete.
Consistent meant that the statement would not allow any inconsistencies of any kind. If you used the logic and the foundantions of the Principia, you couldn’t even say anything like Russell’s paradox: you couldn’t even write it as a valid statement.
Complete meant that every true statement was provably true, every false statement was provably false, and every statement was either true or false.
A big part of how it did this was by creating a very strict stratification of logic – the stratification that we discussed in ST type theory. The principia’s type-theoretic logic could reason about a specific number, using level-0 statements. You could reason about sets of numbers using level-1 statements. And you could reason about sets of sets of numbers using level-2 statements. In a level-1 statement, you could make meta-statements about level-0 properties, but you couldn’t reason about level-1. In level-2, you could reason about level-1 and level-0, but not about level-2. This meant that you couldn’t make a set like the set of all sets that don’t contain themselves – because that set is formed using a predicate – and when you’re forming a set like “the set of all sets that don’t contain themselves”, the sets you can reason about are a level below you. You can form a level-1 set using a level-1 statement about level-0 objects. But you can’t make a level-1 statement about level-1 objects! So self-referential systems would be completely impossible in the Principia’s logic. You just can’t even state Russell’s paradox. It’s inexpressible.
As a modern student of math, it’s hard to understand what a profound thing they were trying to do. We’ve grown up learning math long after incompleteness became a well-known fact of life. (I read “Gödel Escher Bach” when I was a college freshman – well before I took any particularly deep math classes – so I knew about incompleteness before I knew enough to really understand what completeness woud have meant!) The principia would have been the perfection of math, a final ultimate perfect system. There would have been nothing that we couldn’t prove, nothing in math that we couldn’t know!
What Gödel did was show that using the Principia’s own system, and it’s own syntax, that not only was the principia itself flawed, but that any possible effort like the principia would inevitably be flawed!
With the incompleteness proof, Gödel showed that even in the Principia, even with all of the effort that it made to strictly separate the levels of reasoning, that he could form self-referential statements, and that those self-referential statements were both true and unprovable.
The way that he did it was simply brilliant. The proof was a sequence of steps.
He showed that using Peano arithmetic – that is, the basic definition of natural numbers and natural number arithmetic – that you could take any principia-logic statement, and uniquely encode it as a number – so that every logical statement was a number, and ever number was a specific logical statement.
Then using that basic mechanic, he showed how you could take any property defined by a predicate in the principia’s logic, and encode it as a arithmetic property of the numbers. So a number encoded a statement, and the property of a number could be encoded arithmetically. A number, then, could be both a statement, and a definition of an arithmetic property of a stament, and a logical description of a logical property of a statement – all at once!
Using that encoding, then – which can be formed for any logic that can express Peano arithmetic – he showed that you could form a self-referential statement: a number that was a statement about numbers including the number that was statement itself. And more, it could encode a meta-property of the statement in a way that was both true, and also unprovable: he showed how to create a logical property “There is no proof of this statement”, which applied to its own numeric encoding. So the statement said, about itself, that it couldn’t be proven.
The existence of that statement meant that the Principia – and any similar system! – was incomplete. Completeness means that every true statement is provably true within the system. But the statement encodes the fact that it cannot be proven. If you could prove it, the system would be inconsistent. If you can’t, it’s consistent, but incomplete.
Finally, we’re at the end of our walkthrough of Gödel great incompleteness proof. As a refresher, the basic proof sketch is:
Take a simple logic. We’ve been using a variant of the Principia Mathematica’s logic, because that’s what GÃ¶del used.
Show that any statement in the logic can be encoded as a number using an arithmetic process based on the syntax of the logic. The process of encoding statements numerically is called GÃ¶del numbering.
Show that you can express meta-mathematical properties of logical statements in terms of arithemetic properties of their GÃ¶del numbers. In particular, we need to build up the logical infrastructure that we need to talk about whether or not a statement is provable.
Using meta-mathematical properties, show how you can create an unprovable statement encoded as a GÃ¶del number.
What came before:
Gödel numbering: The logic of the Principia, and how to encode it as numbers. This was step 1 in the sketch.
Arithmetic Properties: what it means to say that a property can be expressed arithemetically. This set the groundwork for step 2 in the proof sketch.
Encoding meta-math arithmetically: how to take meta-mathematical properties of logical statements, and define them as arithmetic properties of the Gödel numberings of the statements. This was step 2 proper.
So now we can move on to step three, where we actually see why mathematical logic is necessarily incomplete.
When we left off with Gödel, we’d gone through a very laborious process showing how we could express meta-mathematical properties of logical statements as primitive recursive functions and relations. We built up to being able to express one non-primitive recursive property, which describes the property that a given statement is provable:
pred provable(x) =
some y {
proofFor(y, x)
}
}
The reason for going through all of that was that we really needed to show how we could capture all of the necessary properties of logical statements in terms of arithmetic properties of their Gödel numbers.
Now we can get to the target of Gödel’s effort. What Gödel was trying to do was show how to defeat the careful stratification of the Principia’s logic. In the principia, Russell and Whitehead had tried to avoid problems with self-reference by creating a very strict stratification, where each variable or predicate had a numeric level, and could only reason about objects from lower levels. So if natural numbers were the primitive objects in the domain being reasoned about, then level-1 objects would be things like specific natural numbers, and level-1 predicates could reason about specific natural numbers, but not about sets of natural numbers or predicates over the natural numbers. Level-2 objects would be sets of natural numbers, and level-2 predicates could reason about natural numbers and sets of natural numbers, but not about predicates over sets of natural numbers, or sets of sets of natural numbers. Level-3 objects would be sets of sets of natural numbers… and so on.
The point of this stratification was to make self-reference impossible. You couldn’t make a statement of the form “This predicate is true”: the predicate would be a level-N predicate, and only a level N+1 predicate could reason about a level-N predicate.
What Gödel did in the laborious process we went through in the last post is embed a model of logical statements in the natural numbers. That’s the real trick: the logic is designed to work with a set of objects that are a model of the natural numbers. By embedding a model of logical statements in the natural numbers, he made it possible for a level-1 predicate (a predicate about a specific natural number) to reason about any logical statement or object. A level-1 predicate can now reason about a level-7 object! A level-1 predicate can reason about the set defined by a level-1 predicate: a level-1 predicate can reason about itself!.
Now, we can finally start getting to the point of all of this: incompleteness! We’re going to use our newfound ability to nest logical statements into numbers to construct an unprovable true statement.
In the last post, one of the meta-mathematical properties that we defined for the Gödel-numbered logic was immConseq, which defines when some statement x is an immediate consequence of a set of statements S. As a reminder, that means that x can be inferred from statements in S in one inferrence step.
We can use that property to define what it means to be a consequence of a set of statements: it’s the closure of immediate consequence. We can define it in pseudo-code as:
def conseq(κ) = {
K = κ + axioms
added_to_k = false
do {
added_to_k = false
for all c in immConseq(K) {
if c not in K {
add c to K
added_to_k = true
}
}
} while added_to_k
return K
}
In other words, Conseq(κ) is the complete set of everything that can possibly be inferred from the statements in κ and the axioms of the system. We can say that there’s a proof for a statement x in κ if and only if x ∈ Conseq(κ).
We can the idea of Conseq use that to define a strong version of what it means for a logical system with a set of facts to be consistent. A system is ω-consistent if and only if there is not a statement a such that: a ∈ Conseq(κ) ∧ not(forall(v, a)) ∈ Conseq(κ).
In other words, the system is ω-consistent as long as it’s never true that both a universal statement and it. But for our purposes, we can treat it as being pretty much the same thing. (Yes, that’s a bit hand-wavy, but I’m not trying to write an entire book about Gödel here!)
(Gödel’s version of the definition of ω-consistency is harder to read than this, because he’s very explicit about the fact that Conseq is a property of the numbers. I’m willing to fuzz that, because we’ve shown that the statements and the numbers are interchangable.)
Using the definition of ω-consistency, we can finally get to the actual statement of the incompleteness theorem!
Gödel’s First Incompleteness Theorem: For every ω-consistent primitive recursive set κ of formulae, there is a primitive-recursive predicate r(x) such that neither forall(v, r) nor not(forall(v, r)) is provable.
To prove that, we’ll construct the predicate r.
First, we need to define a version of our earlier isProofFigure that’s specific to the set of statements κ:
pred isProofFigureWithKappa(x, kappa) = {
all n in 1 to length(x) {
isAxiom(item(n, x)) or
item(n, x) in kappa or
some p in 0 to n {
some q in 0 to n {
immedConseq(item(n, x), item(p, x), item(q, x))
}
}
} and length(x) > 0
}
This is the same as the earlier definition – just specialized so that it ensures that every statement in the proof figure is either an axiom, or a member of κ.
We can do the same thing to specialize the predicate proofFor and provable:
pred proofForStatementWithKappa(x, y, kappa) = {
isProofFigureWithKappa(x, kappa) and
item(length(x), x) = y
}
pred provableWithKappa(x, kappa) = {
some y {
proofForStatementWithKappa(y, x, kappa)
}
}
If κ is the set of basic truths that we can work with, then provable in κ is equivalent to provable.
Now, we can define a predicate UnprovableInKappa:
pred NotAProofWithKappa(x, y, kappa) = {
not (proofForKappa(x, subst(y, 19, number(y))))
}
Based on everything that we’ve done so far, NotAProofWithKappa is primitive recursive.
This is tricky, but it’s really important. We’re getting very close to the goal, and it’s subtle, so let’s take the time to understand this.
Remember that in a Gödel numbering, each prime number is a variable. So 19 here is just the name of a free variable in y.
Using the Principia’s logic, the fact that variable 19 is free means that the statement is parametric in variable 19. For the moment, it’s an incomplete statement, because it’s got an unbound parameter.
What we’re doing in NotAProofWithKappa is substituting the numeric coding of y for the value of y‘s parameter. When that’s done, y is no longer incomplete: it’s unbound variable has been replaced by a binding.
With that substitution, NotAProofWithKappa(x, y, kappa) is true when xdoes not prove that y(y) is true.
What NotAProofWithKappa does is give us a way to check whether a specific sequence of statements x is not a proof of y.
We want to expand NotAProofWithKappa to something universal. Instead of just saying that a specific sequence of statements x isn’t a proof for y, we want to be able to say that no possible sequence of statements is a proof for y. That’s easy to do in logic: you just wrap the statement in a “∀ x ( )”. In Gödel numbering, we defined a function that does exactly that. So the universal form of provability is: ∀ a (NotAProofWithKappa(a, y, kappa)).
In terms of the Gödel numbering, if we assume that the Gödel number for the variable a is 17, and the variable y is numbered as 19, we’re talking about the statement p = forall(17, ProvableInKappa(17, 19, kappa).
p is the statement that for some logical statement (the value of variable 19, or y in our definition), there is no possible value for variable 17 (a) where a proves y in κ.
All we need to do now is show that we can make p become self-referential. No problem: we can just put number(p) in as the value of y in UnprovableInKappa. If we let q be the numeric value of the statement UnprovableInKappa(a, y), then:
r = subst(q, 19, p)
i = subst(p, 19, r)
i says that there is no possible value x that proves p(p). In other words, p(p) is unprovable: there exists no possible proof that there is no possible proof of p!
This is what we’ve been trying to get at all this time: self-reference! We’ve got a predicate y which is able to express a property of itself. Worse, it’s able to express a negative property of itself!
Now we’re faced with two possible choices. Either i is provable – in which case, κ is inconsistent! Or else i is unprovable – in which case κ is incomplete, because we’ve identified a true statement that can’t be proven!
That’s it: we’ve shown that in the principia’s logic, using nothing but arithmetic, we can create a true statement that cannot be proven. If, somehow, it were to be proven, the entire logic would be inconsistent. So the principia’s logic is incomplete: there are true statements that cannot be proven true.
We can go a bit further: the process that we used to produce this result about the Principia’s logic is actually applicable to other logics. There’s no magic here: if your logic is powerful enough to do Peano arithmetic, you can use the same trick that we demonstrated here, and show that the logic must be either incomplete or inconsistent. (Gödel proved this formally, but we’ll just handwave it.)
Looking at this with modern eyes, it doesn’t seem quite as profound as it did back in Gödel’s day.
When we look at it through the lens of today, what we see is that in the Principia’s logic, proof is a mechanical process: a computation. If every true statement was provable, then you could take any statement S, and write a program to search for a proof of either S or ¬ S, and eventually, that program would find one or the other, and stop.
In short, you’d be able to solve the halting problem. The proof of the halting problem is really an amazingly profound thing: on a very deep level, it’s the same thing as incompleteness, only it’s easier to understand.
But at the time that Gödel was working, Turing hadn’t written his paper about the halting problem. Incompletess was published in 1931; Turing’s halting paper was published in 1936. This was a totally unprecedented idea when it was published. Gödel produced one of the most profound and surprising results in the entire history of mathematics, showing that the efforts of the best mathematicians in the world to produce the perfection of mathematics were completely futile.
As you may be figuring out, there’s a reason why I resisted walking through Gödel’s proof of incompleteness for so long. Incompeteness isn’t a simple proof!
To refresh your memory, here’s a sketch of the proof:
Take a simple logic. We’ve been using a variant of the Principia Mathematica’s logic, because that’s what Gödel used.
Show that any statement in the logic can be encoded as a number using an arithmetic process based on the syntax of the logic. The process of encoding statements numerically is called Gödel numbering.
Show that you can express meta-mathematical properties of logical statements in terms of arithemetic properties of their Gödel numbers. In particular, we need to build up the logical infrastructure that we need to talk about whether or not a statement is provable.
Using meta-mathematical properties, show how you can create an unprovable statement encoded as a Gödel number.
What we’ve done so far is the first two steps, and part of the third. In this post, we saw the form of the Principia’s logic that we’re using, and how to numerically encode it as a Gödel numbering. We’ve start started on the third point in this post, by figuring out just what it means to say that things are encoded arithmetically. Now we can get to the part where we see how to encode meta-mathematical properties in terms of arithmetic properties of the Gödel numbering. In this post, we’re going to build up everything we need to express syntactic correctness, logical validity, and provability in terms of arithmetical properties of Gödel numbers. (And, as a reminder, I’ve been using this translation on Gödel’s original paper on incompleteness.)
This is the most complex part of the incompleteness proof. The basic concept of what we’re doing is simple, but the mechanics are very difficult. What we want to do is define a set of predicates about logical statements, but we want those predicates to be expressed as arithmetic properties of the numerical representations of the logical statements.
The point of this is that we’re showing that done in the right way, arithmetic is logic – that doing arithmetic on the Gödel numbers is doing logical inference. So what we need to do is build up a toolkit that shows us how to understand and manipulate logic-as-numbers using arithmetic. As we saw in the last post, primitive recursion is equivalent to arithmetic – so if we can show how all of the properties/predicates that we define are primitive recursive, then they’re arithmetic.
This process involves a lot of steps, each of which is building the platform for the steps that follow it. I struggled quite a bit figuring out how to present these things in a comprehensible way. What I ended up with is writing them out as code in a pseudo-computer language. Before inventing this language, I tried writing actual executable code, first in Python and then in Haskell, but I wasn’t happy with the clarity of either one.
Doing it in an unimplemented language isn’t as big a problem as you might think. Even if this was all executable, you’re not going to be able to actually run any of it on anything real – at least not before you hair turns good and gray. The way that this stuff is put together is not what any sane person would call efficient. But the point isn’t to be efficient: it’s to show that this is possible. This code is really all about searching; if we wanted to be efficient, this could all be done in a different representation, with a different search method that was a lot faster – but that wolud be harder to understand.
So, in the end, I threw together a simple language that’s easy to read. This language, if it were implemented, wouldn’t really even be Turing complete – it’s a primitive recursive language.
Basics
We’ll start off with simple numeric properties that have no obvious connection to the kinds of meta-mathematical statements that we want to talk about, but we’ll use those to define progressively more and more complex and profound properties, until we finally get to our goal.
# divides n x == True if n divides x without remainder.
pred divides(n, x) = x mod n == 0
pred isPrime(0) = False
pred isPrime(1) = False
pred isPrime(2) = True
pred isPrime(n) = {
all i in 2 to n {
not divides(i, n)
}
}
fun fact(0) = 1
fun fact(n) = n * fact(n - 1)
Almost everything we’re going to do here is built on a common idiom. For anything we want to do arithmetically, we’re going to find a bound – a maximum numeric value for it. Then we’re going to iterate over all of the values smaller than that bound, searching for our target.
For example, what’s the nth prime factor of x? Obviously, it’s got to be smaller than x, so we’ll use x as our bound. (A better bound would be the square root of x, but it doesn’t matter. We don’t care about efficiency!) To find the nth prime factor, we’ll iterate over all of the numbers smaller than our bound x, and search for the smallest number which is prime, which divides x, and which is larger than the n-1th prime factor of x. We’ll translate that into pseudo-code:
fun prFactor(0, x) = 0
fun prFactor(n, x) = {
first y in 1 to x {
isPrime(y) and divides(y, x) and prFactor(n - 1, x) < y
}
}
Similarly, for extracting values from strings, we need to be able to ask, in general, what's the nth prime number? This is nearly identical to prFactor above. The only difference is that we need a different bound. Fortunately, we know that the nth prime number can't be larger than the factorial of the previous prime plus 1.
fun nthPrime(0) = 0
fun nthPrime(n) = {
first y in 1 to fact(nthPrime(n - 1)) + 1 {
isPrime(y) and y > nthPrime(n - 1))
}
}
In composing strings of Gödel numbers, we use exponentiation. Given integers x and n, x^{n}, we can obviously compute them via primitive recursion. I'll define them below, but in the rest of this post, I'll write them as an operator in the language:
fun pow(n, 0) = 1
fun pow(n, i) = n * pow(n, i - 1)
String Composition and Decomposition
With those preliminaries out of the way, we can get to the point of defining something that's actually about one of the strings encoded in these Gödel numbers. Given a number n encoding a string, item(n, x) is the value of the nth character of x. (This is slow. This is really slow! We're getting to the limit of what a very powerful computer can do in a reasonable amount of time. But this doesn't matter. The point isn't that this is a good way of doing these things: it's that these things are possible. To give you an idea of just how slow this is, I started off writing the stuff in this post in Haskell. Compiled with GHC, which is a very good compiler, item to extract the 6th character of an 8 character string took around 10 minutes on a 2.4Ghz laptop.)
fun item(n, x) = {
first y in 1 to x {
divides(prFactor(n, x) ** y, y) and
not divides(prFactor(n, x)**(y+1), x)
}
}
Given a string, we want to be able to ask how long it is; and given two strings, we want to be able to concatenate them.
fun length(x) = {
first y in 1 to x {
prFactor(y, x) > 0 and prFactor(y + 1, x) == 0
}
}
fun concat(x, y) = {
val lx = length(x)
val ly = length(y)
first z in 1 to nthprime(lx + ly)**(x + y) {
(all n in 1 to lx {
item(n, z) == item(n, x)
}) and (all n in 1 to ly {
item(n + lx, z) == item(n, y)
})
}
}
fun concatl([]) = 0
fun concatl(xs) = {
concat(head(xs), concatl(tail(xs)))
}
fun seq(x) = 2**x
We want to be able to build statements represented as numbers from other statements represented as numbers. We'll define a set of functions that either compose new strings from other strings, and to check if a particular string is a particular kind of syntactic element.
# x is a variable of type n.
pred vtype(n, x) = {
some z in 17 to x {
isPrime(z) and x == n**z
}
}
# x is a variable
pred isVar(x) = {
some n in 1 to x {
vtype(n, x)
}
}
fun paren(x) =
concatl([gseq(11), x, gseq(13)])
# given the Gödel number for a statement x, find
# the Gödel number for not x.
fun gnot(x) =
concat(gseq(5), paren(x))
# Create the number for x or y.
fun gor(x, y) =
concatl([paren(x), seq(7), paren(y)])
# Create the number for 'forall x(y)'.
fun gforall(x, y) =
concatl([seq(9), seq(x), paren(y)])
# Create the number for x with n invocations of the primitive
# successor function.
fun succn(0, x) = x
fun succn(n, x) = concat(seq(3), succn(n - 1, x))
# Create the number n using successor and 0.
fun gnumber(n) = succn(n, seq(1))
# Check if a statement is type-1.
pred stype_one(x) = {
some m in 1 to x {
m == 1 or (vtype(1, m) and x == succn(n, seq(m))
}
}
# Check if a statement is type n.
pred fstype(1, x) = stype_one(x)
pred fstype(n, x) =
some v in 1 to x {
vtype(n, v) and R(v)
}
}
That last function contains an error: the translation of Gödel that I'm using says R(v) without defining R. Either I'm missing something, or the translator made an error.
Formulae
Using what we've defined so far, we're now ready to start defining formulae in the basic Principia logic. Forumlae are strings, but they're strings with a constrained syntax.
pred elFm(x) = {
some y in 1 to x {
some z in 1 to x {
some n in 1 to x {
stype(n, y) and stype(n+1, z) and x == concat(z, paren(y))
}
}
}
}
All this is doing is expressing the grammar rule in arithmetic form: an elementary formula is a predicate: P(x), where x is a variable on level n, and P is a variable of level x + 1.
The next grammar rule that we encode this way says how we can combine elementary formulae using operators. There are three operators: negation, conjunction, and universal quantification.
pred op(x, y, z) = {
x == gnot(y) or
x == gor(y, z) or
(some v in 1 to x { isVar(v) and x == gforall(v, y) })
}
And now we can start getting complex. We're going to define the idea of a valid sequence of formulae. x is a valid sequence of formulae when it's formed from a collection of formulae, each of which is either an elementary formula, or is produced from the formulae which occured before it in the sequence using either negation, logical-or, or universal quantification.
In terms of a more modern way of talking about it, the syntax of the logic is a grammar. A formula sequence, in this system, is another way of writing the parse-tree of a statement: the sequence is the parse-tree of the last statement in the sequence.
pred fmSeq(x) = {
all p in 0 to length(x) {
elFm(item(n, x)) or
some p in 0 to (n - 1) {
some q in 0 to (n - 1) {
op(item(n,x), item(p, x), item(q, x))
}
}
}
}
The next one bugs me, because it seems wrong, but it isn't really! It's a way of encoding the fact that a formula is the result of a well-defined sequence of formulae. In order to ensure that we're doing primitive recursive formulae, we're always thinking about sequences of formulae, where the later formulae are produced from the earlier ones. The goal of the sequence of formula is to produce the last formula in the sequence. What this predicate is really saying is that a formula is a valid formula if there is some sequence of formulae where this is the last one in the sequence.
Rephrasing that in grammatical terms, a string is a formula if there is valid parse tree for the grammar that produces the string.
pred isFm(x) = {
some n in 1 to nthPrime(length(x)**2)**(x*length(x)**2) {
fmSeq(n)
}
}
So, now, can we say that a statement is valid because it's parsed according to the grammar? Not quite. It's actually a familiar problem for people who write compilers. When you parse a program in some language, the grammar doesn't usually specify variables must be declared before they're used. It's too hard to get that into the grammar. In this logic, we've got almost the same problem: the grammar hasn't restricted us to only use bound variables. So we need to have ways to check whether a variable is bound in a Gödel-encoded formula, and then use that to check the validity of the formula.
# The variable v is bound in formula x at position n.
pred bound(v, n, x) = {
isVar(v) and isFm(x) and
(some a in 1 to x {
some b in 1 to x {
some c in 1 to x {
x == concatl([a, gforall(v, b), c]) and
isFm(b) and
length(a) + 1 ≤ n ≤ length(a) + length(forall(v, b))
}
}
})
}
# The variable v in free in formula x at position n
pred free(v, n, x) = {
isVar(v) and isFm(x) and
(some a in 1 to x {
some b in 1 to x {
some c in 1 to x {
v == item(n, x) and n ≤ length(x) and not bound(v, n, x)
}
}
})
}
pred free(v, x) = {
some n in 1 to length(x) {
free(v, n, x)
}
}
To do logical inference, we need to be able to do things like replace a variable with a specific infered value. We'll define how to do that:
# replace the item at position n in x with y.
fun insert(x, n, y) = {
first z in 1 to nthPrime(length(x) + length(y))**(x+y) {
some u in 1 to x {
some v in 1 to x {
x == concatl([u, seq(item(n, x)), v]) and
z == concatl([u, y, v]) and
n == length(u) + 1
}
}
}
}
There are inference operations and validity checks that we can only do if we know whether a particular variable is free at a particular position.
# freePlace(k, v, k) is the k+1st place in x (counting from the end)
# where v is free.
fun freePlace(0, v, x) = {
first n in 1 to length(x) {
free(v, n, x) and
not some p in n to length(x) {
free(v, p, x)
}
}
}
fun freePlace(k, v, x) = {
first n in 1 to freePlace(n, k - 1, v) {
free(v, n, x) and
not some p in n to freePlace(n, k - 1, v) {
free(v, p, x)
}
}
}
# number of places where v is free in x
fun nFreePlaces(v, x) = {
first n in 1 to length(x) {
freeplace(n, v, x) == 0
}
}
In the original logic, some inference rules are defined in terms of a primitive substitution operator, which we wrote as subst[v/c](a) to mean substitute the value c for the variable c in the statement a. We'll build that up on a couple of steps, using the freePlaces function that we just defined.
# Subst1 replaces a single instance of v with y.
fun subst'(0, x, v, y) = x
fun subst1(0k, x, v, y) =
insert(subst1(k, x, v, y), freePlace(k, v, x), y)
# subst replaces all instances of v with y
fun subst(x, v, y) = subst'(nFreePlaces(v, x), x, v, y)
The next thing we're going to do isn't, strictly speaking, absolutely necessary. Some of the harder stuff we want to do will be easier to write using things like implication, which aren't built in primitive of the Principia logic. To write those as clearly as possible, we'll define the full suite of usual logical operators in terms of the primitives.
# implication
fun gimp(x, y) = gor(gnot(x), y)
# logical and
fun gand(x, y) = gnot(gor(gnot(x), gnot(y)))
# if/f
fun gequiv(x, y) = gand(gimp(x, y), gimp(y, x))
# existential quantification
fun gexists(v, y) = not(gforall(v, not(y)))
Axioms
The Peano axioms are valid logical statements, so they have Gödel numbers in this system. We could compute their value, but why bother? We know that they exist, so we'll just give them names, and define a predicate to check if a value matches them.
The form of the Peano axioms used in incompleteness are:
Zero: ¬(succ(x_{1}) = 0)
Uniqueness: succ(x_{1}) = succ(y_{1}) Rightarrow x = y
const pa1 = ...
const pa2 = ...
const pa3 = ...
pred peanoAxiom(x) =
(x == pa1) or (x == pa2) or (x == pa3)
Similarly, we know that the propositional axioms must have numbers. The propositional
axioms are:
I'll show the translation of the first - the rest follow the same pattern.
# Check if x is a statement that is a form of propositional
# axiom 1: y or y => y
pred prop1Axiom(x) =
some y in 1 to x {
isFm(x) and x == imp(or(y, y), y)
}
}
pred prop2Axiom(x) = ...
pred prop3Axiom(x) = ...
pred prop4Axiom(x) = ...
pred propAxiom(x) = prop2Axiom(x) or prop2Axiom(x) or
prop3Axiom(x) or prop4Axiom(x)
Similarly, all of the other axioms are written out in the same way, and we add a predicate isAxiom to check if something is an axiom. Next is quantifier axioms, which are complicated, so I'll only write out one of them - the other follows the same basic scheme.
The two quantifier axioms are:
quantifier_axiom1_condition(z, y, v) = {
not some n in 1 to length(y) {
some m in 1 to length(z) {
some w in 1 to z {
w == item(m, z) and bound(w, n, y) and free(v, n, y)
}
}
}
}
pred quantifier1Axiom(x) = {
some v in 1 to x {
some y in 1 to x {
some z in 1 to x {
some n in 1 to x {
vtype(n, v) and stype(n, z) and
isFm(y) and
quantifier_axiom1_condition(z, y, v) and
x = gimp(gforall(v, y), subst(y, v, z))
}
}
}
}
}
quanitifier_axiom2 = ...
isQuantifierAxiom = quantifier1Axiom(x) or quantifier2Axiom(x)
We need to define a predicate for the reducibility axiom (basically, the Principia's version of the ZFC axiom of comprehension). The reducibility axiom is a schema: for any predicate , . In our primitive recursive system, we can check if something is an instance of the reducibility axiom schema with:
pred reduAxiom(x) =
some u in 1 to x {
some v in 1 to x {
some y in 1 to x {
some n in 1 to x {
vtype(n, v) and
vtype(n+1, u) and
not free(u, y) and
isFm(y) and
x = gexists(u, gforall(v, gequiv(concat(seq(u), paren(seq(v))), y)))
}
}
}
}
}
Now, the set axiom. In the logic we're using, this is the axiom that defines set equality. It's written as . Set equality is defined for all types of sets, so we need to have one version of axiom for each level. We do that using type-lifting: we say that the axiom is true for type-1 sets, and that any type-lift of the level-1 set axiom is also a version of the set axiom.
fun typeLift(n, x) = {
first y in 1 to x**(x**n) {
all k in 1 to length(x) {
item(k, x) ≤ 13 and item(k, y) == item(k, v) or
item(k, x) > 13 and item(k, y) = item(k, x) * prFactor(1, item(k, x))**n
}
}
}
We haven't defined the type-1 set axiom. But we just saw the axiom above, and it's obviously a simple logical statement. That mean that it's got a Gödel number. Instead of computing it, we'll just say that that number is called sa1. Now we can define a predicate to check if something is a set axiom:
val sa1 = ...
pred setAxiom(x) =
some n in 1 to x {
x = typeLift(n, sa)
}
}
We've now defined all of the axioms of the logic, so we can now create a general predicate to see if a statement fits into any of the axiom categories:
pred isAxiom(x) =
peanoAxiom(x) or propAxiom(x) or quantifierAxom(x) or
reduAxiom(x) or setAxiom(x)
Proofs and Provability!
With all of the axioms expressible in primitive recursive terms, we can start on what it means for something to be provable. First, we'll define what it means for some statement x to be an immediate consequence of some statements y and z. (Back when we talked about the Principia's logic, we said that x is an immediate consequence of y and z if either: y is the formula z ⇒ x, or if c is the formula ∀v.x).
pred immConseq(x, y, z) = {
y = imp(z, x) or
some v in 1 to x {
isVar(v) and x = forall(v, y)
}
}
Now, we can use our definition of an immediate consequence to specify when a sequence of formula is a proof figure. A proof figure is a sequence of statements where each statement in it is either an axiom, or an immediate consequence of two of the statements that preceeded it.
pred isProofFigure(x) = {
(all n in 0 to length(x) {
isAxiom(item(n, x)) or
some p in 0 to n {
some q in 0 to n {
immConseq(item(n, x), item(p, x), item(q, x))
}
}
}) and
length(x) > 0
}
We can say that x is a proof of y if x is proof figure, and the last statement in x is y.
pred proofFor(x, y) =
isProofFigure(x) and
item(length(x), x) == y
Finally, we can get to the most important thing! We can define what it means for something to be provable! It's provable if there's a proof for it!
pre provable(x) =
some y {
proofFor(y, x)
}
}
Note that this last one is not primitive recursive! There's no way that we can create a bound for this: a proof can be any length.
At last, we're done with these definition. What we've done here is really amazing: now, every logical statement can be encoded as a number. Every proof in the logic can be encoded as a sequence of numbers: if something is provable in the Principia logic, we can encode that proof as a string of numbers, and check the proof for correctness using nothing but (a whole heck of a lot of) arithmetic!
Next post, we'll finally get to the most important part of what Gödel did. We've been able to define what it means for a statement to be provable - we'll use that to show that there's a way of creating a number encoding the statement that something is not provable. And we'll show how that means that there is a true statement in the Principia's logic which isn't provable using the Principia's logic, which means that the logic isn't complete.
In fact, the proof that we'll do shows a bit more than that. It doesn't just show that the Principia's logic is incomplete. It shows that any consistent formal system like the Principia, any system which is powerful enough to encode Peano arithmetic, must be incomplete.
When I left off, we’d seen how to take statements written in the logic of the Principia Mathematica, and convert them into numerical form. What we need to see now is how to get meta-mathematical.
We want to be able to write second-order logical statements. The basic trick to incompleteness is that we’re going to use the numerical encoding of statements to say that a predicate or relation is represented by a number. Then we’re going to write predicates about predicates by defining predicates on the numerical representations of the first-order predicates. That’s going to let us create a true statement in the logic that can’t be proven with the logic.
To do that, we need to figure out how to take our statements and relations represented as numbers, and express properties of those statements and relations in terms of arithmetic. To do that, we need to define just what it means to express something arithmetically. Gödel did that by defining “arithmetically” in terms of a concept called primitive recursion.
I learned about primitive recursion when I studied computational complexity. Nowadays, it’s seen as part of theoretical computer science. The idea, as we express it in modern terms, is that there are many different classes of computable functions. Primitive recursion is one of the basic complexity classes. You don’t need a Turing machine to compute primitive recursive functions – they’re a simpler class.
The easiest way to understand primitive recursion is that it’s what you get in a programming language with integer arithmetic, and simple for-loops. The only way you can iterate is by repeating things a bounded number of times. Primitive recursion has a lot of interesting properties: the two key ones for our purposes here are: number theoretic proofs are primitive recursive, and every computation of a primitive recursive function is guaranteed to complete within a bounded amount of time.
The formal definition of primitive recursion, the way that Gödel wrote it, is quite a bit more complex than that. But it means the same thing.
We start with what it means to define a formula via primitive recursion. (Note the language that I used there: I’m not explaining what it means for a function to be primitive recursive; I’m explaining what it means to be defined via primitive recursion.) And I’m defining formulae, not functions. In Gödel’s proof, we’re always focused on numerical reasoning, so we’re not going to talk about programs or algorithms, we’re going to about the definition of formulae.
A formula is defined via primitive recursion if, for some other formulae and :
Base:
Recursive: .
So, basically, the first parameter is a bound on the number of times that can invoked recursively. When it’s 0, you can’t invoke any more.
A formula is primitive recursive if it defined from a collection of formulae where any formula is defined via primitive recursion from , or the primitive succ function from Peano arithmetic.
For any formula in that sequence, the degree of the formula is the number of other primitive recursive formulae used in its definition.
Now, we can define a primitive recursive property: is primitive recursive if and only if there exists a primitive recursive function such that .
With primitive recursive formulae and relations defined, there’s a bunch of theorems about how you can compose primitive recursive formulae and relations:
Every function or relation that you get by substituting a primitive recursive function for a variable in a primitive recursive function/relation is primitive recursive.
If R and S are primitive relations, then ¬R, R∧S, R∨S are all primitive recursive.
If and are primitive recursive functions, then the relation is also primitive recursive.
Let and be finite-length tuples of variables. If the function and the relation are primitive recursive, then so are the relations:
Let and be finite-length tuples of variables. And let be the smallest value of for which and is true, or 0 if there is no such value. Then if the function and the relation are primitive recursive, then so is the function .
By these definitions, addition, subtraction, multiplication, and integer division are all primitive recursive.
Ok. So, now we’ve got all of that out of the way. It’s painful, but it’s important. What we’ve done is come up with a good formal description of what it means for something to be an arithmetic property: if we can write it as a primitive recursive relation or formula, it’s arithmetic.
So now, finally, we’re ready to get to the really good part! Now that we know what it means to define something arithmetically, we can see how to define meta-mathematical properties and formulae arithmetically. Next post, hopefully tomorrow, I’ll start showing you arithmetic expressions of meta-math!
The first step in Gödel’s incompleteness proof was finding a way of taking logical statements and encoding them numerically. Looking at this today, it seems sort-of obvious. I mean, I’m writing this stuff down in a text file – that text file is a stream of numbers, and it’s trivial to convert that stream of numbers into a single number. But when Gödel was doing it, it wasn’t so obvious. So he created a really clever mechanism for numerical encoding. The advantage of Gödel’s encoding is that it makes it much easier to express properties of the encoded statements numerically.
Before we can look at how Gödel encoded his logic into numbers, we need to look at the logic that he used. Gödel worked with the specific logic variant used by the Principia Mathematica. The Principia logic is minimal and a bit cryptic, but it was built for a specific purpose: to have a minimal syntax, and a complete but minimal set of axioms.
The whole idea of the Principia logic is to be purely syntactic. The logic is expected to have a valid model, but you shouldn’t need to know anything about the model to use the logic. Russell and Whitehead were deliberately building a pure logic where you didn’t need to know what anything meant to use it. I’d really like to use Gödel’s exact syntax – I think it’s an interestingly different way of writing logic – but I’m working from a translation, and the translator updated the syntax. I’m afraid that switching between the older Gödel syntax, and the more modern syntax from the translation would just lead to errors and confusion. So I’m going to stick with the translation’s modernization of the syntax.
The basic building blocks of the logic are variables. Already this is a bit different from what you’re probably used to in a logic. When we think of logic, we usually consider predicates to be a fundamental thing. In this logic, they’re not. A predicate is just a shorthand for a set, and a set is represented by a variable.
Variables are stratified. Again, it helps to remember where Russell and Whitehead were coming from when they were writing the Principia. One of their basic motivations was avoiding self-referential statements like Russell’s paradox. In order to prevent that, they thought that they could create a stratified logic, where on each level, you could only reason about objects from the level below. A first-order predicate would be a second-level object could only reason about first level objects. A second-order predicate would be a third-level object which could reason about second-level objects. No predicate could ever reason about itself or anything on its on level. This leveling property is a fundamental property built into their logic. The way the levels work is:
Type one variables, which range over simple atomic values, like specific single natural numbers. Type-1 variables are written as a_{1}, b_{1}.
Type two variables, which range over sets of atomic values, like sets of natural numbers. A predicate, like IsOdd, about specific natural numbers would be represented as a type-2 variable. Type-2 variables are written a_{2}, b_{2}, …
Type three variables range over sets of sets of atomic values. The mappings of a function could be represented as type-3 variables: in set theoretic terms, a function is set of ordered pairs. Ordered pairs, in turn, can be represented as sets of sets – for example, the ordered pair (1, 4) would be represented by the set { {1}, {1, 4} }. A function, in turn, would be represented by a type-4 variable – a set of ordered pairs, which is a set of sets of sets of values.
Using variables, we can form simple atomic expressions, which in Gödel’s terminology are called signs. As with variables, the signs are divided into stratified levels:
Type-1 signs are variables, and successor expressions. Successor expressions are just Peano numbers written with “succ”: 0, succ(0), succ(succ(0)), succ(a_{1}), etc.
Signs of any type greater than 1 are just variables of that type/level.
Now we can assemble the basic signs into formulae. Gödel explained how to build formulae in a classic logicians form, which I think is hard to follow, so I’ve converted it into a grammar:
elementary_formula → sign_{n+1}(sign_{n})
formula → ¬(elementary_formula)
formula → (elementary_formula) or (elementary_formula)
formula → ∀ sign_{n} (elementary_formula)
That’s all that’s really included in Gödel’s logic. It’s enough: everything else can be defined in terms of combinatinos of these. For example, you can define logical and in terms of negation and logical or: (a ∧ b) ⇔ ¬ (¬ a ∨ ¬ b).
Next, we need to define the axioms of the system. In terms of logic the way I think of it, these axioms include both “true” axioms, and the inference rules defining how the logic works. There are five families of axioms.
<
ol>
First, there’s the Peano axioms, which define the natural numbers.
¬(succ(x_{1}) = 0): 0 is a natural number, and it’s not the successor of anything.
succ(x_{1}) = succ(y_{1}) ⇒ x_{1} = y_{1}: Successors are unique.
(x_{2}(0) ∧ ∀ x_{1} (x_{2}(x_{1}) ⇒ x_{2}(succ(x_{1})))) ⇒ ∀ x_{1}(x_{2}(x_{1})): induction works on the natural numbers.
Next, we’ve got a set of basic inference rules about simple propositions. These are defined as axiom schemata, which can be instantiated for any set of formalae p, q, and r.
Axioms that define inference over quantification. v is a variable, a is any formula, b is any formula where v is not a free variable, and c is a sign of the same level as v, and which doesn’t have any free variables that would be bound if it were inserted as a replacement for v.
∀ v (a) ⇒ subst[v/c](a): if formula a is true for all values of v, then you can substitute any specific value c for v in a, and a must still be true.
(∀ v (b ∨ a)) ⇒ (b ∨ ∀ v (a))
The Principia’s version of the set theory axiom of comprehension: &exists; u (∀ v ( u(v) ⇔ a )).
<li> And last but not least, an axiom defining set equivalence: <em>∀ x<sub>i</sub> (x<sub>i+1</sub>(x<sub>i</sub>) ⇔ y<sub>i+1</sub>(y<sub>i</sub>)) ⇒ x<sub>i+1</sub> = y<sub>i+1</sub></em> </li>
So, now, finally, we can get to the numbering. This is quite clever. We’re going to use the simplest encoding: for every possible string of symbols in the logic, we’re going to define a representation as a number. So in this representation, we are not going to get the property that every natural number is a valid formula: lots of natural numbers won’t be. They’ll be strings of nonsense symbols. (If we wanted to, we could make every number be a valid formula, by using a parse-tree based numbering, but it’s much easier to just let the numbers be strings of symbols, and then define a predicate over the numbers to identify the ones that are valid formulae.)
We start off by assigning numbers to the constant symbols:
Symbols
Numeric Representation
0
1
succ
3
¬
5
∨
7
∀
9
(
11
)
13
Variables will be represented by powers of prime numbers, for prime numbers greater that 13. For a prime number p, p will represent a type one variable, p^{2} will represent a type two variable, p^{3} will represent a type-3 variable, etc.
Using those symbol encodings, we can take a formula written as symbols x_{1}x_{2}x_{3}…x_{n}, and encode it numerically as the product 2^{x1}3^{x2}5^{x2}…p_{n}^{xn}, where p_{n} is the nth prime number.
For example, suppose I wanted to encode the formula: ∀ x_{1} (y_{2}(x_{1})) ∨ x_{2}(x_{1}).
First, I’d need to encode each symbol:
“∀” would be 9.
“x_{1}“” = 17
“(” = 11
“y_{2}” = 19^{2} = 361
“(” = 11
“x_{1}” = 17
“)” = 13
“∨” = 7
“x_{2}” = 17^{2} = 289
“(” = 11
“x_{1}” = 17
“)” = 13
“)” = 13
The formula would thus be turned into the sequence: [9, 17, 11, 361, 11, 17, 13, 7, 289, 11, 17, 13, 13]. That sequence would then get turned into a single number 2^{9} 3^{17} 5^{11} 7^{361} 11^{11} 13^{17} 17^{13} 19^{7} 23^{289} 29^{11} 31^{17} 37^{13} 41^{13}, which according to Hugs is the number (warning: you need to scroll to see it. a lot!):
Next, we’re going to look at how you can express interesting mathematical properties in terms of numbers. Gödel used a property called primitive recursion as an example, so we’ll walk through a definition of primitive recursion, and show how Gödel expressed primitive recursion numerically.