New Dimensions of Crackpottery

I have, in the past, ranted about how people abuse the word “dimension”, but it’s been a long time. One of my followers on twitter sent me a link to a remarkable piece of crackpottery which is a great example of how people simply do not understand what dimensions are.

There are several ways of defining “dimension” mathematically, but they all come back to one basic concept. A dimension an abstract concept of a direction. We can use the number of dimensions in a space as a way of measuring properties of that space, but those properties all come back to the concept of direction. A dimension is neither a place nor a state of being: it is a direction.

Imagine that you’re sitting in an abstract space. You’re at one point. There’s another point that I want you to go to. In order to uniquely identify your destination, how many directions do I need to mention?

If the space is a line, you only need one: I need to tell you the distance. There’s only one possible direction that you can go, so all I need to tell you is how far. Since you only need one direction, the line is one-dimensional.

If the line is a plane, then I need to tell you two things. I could do that by saying “go right three steps then up 4 steps”, or I could say “turn 53 degrees clockwise, and then walk forward 5 steps.” But there’s no way I can tell you how to get to your destination with less than two directions. You need two directions, so the plane is two dimensional.

If the space is the interior of a cube, then you’ll need three directions, which means that the cube is three dimensional.

On to the crackpottery!

E=mc2 represents a translation across dimensions, from energy to matter.

No, it does not. Energy and matter are not dimensions. e=mc^2 is a statement about the fundamental relation between energy and matter, not a statement about dimensions. Our universe could be 2 dimensional, 3 dimensional, 4 dimensional, or 22 dimensional: relativity would still mean the same thing, and it’s not a statement about a “translation across dimensions”.

Energy can travel at the speed of light, and as Special Relativity tells us, from the perspective of light speed it takes no time to travel any distance. In this way, energy is not bound by time and space the way matter is. Therefore, it is in a way five-dimensional, or beyond time.

Bzzt, no.

Energy does not travel. Light travels, and light can transmit energy, but light isn’t energy. Or, from another perspective, light is energy: but so is everything else. Matter and energy are the same thing.

From the perspective of light speed time most certainly does pass, and it does take plenty of time to travel a distance. Light takes roughly 6 minutes to get from the sun to the earth. What our intrepid author is trying to talk about here is the idea of time dilation. Time dilation describes the behavior of particles with mass when they move at high speeds. As a massive particle moves faster and approaches the speed of light, the mass of the particle increases, and the particle’s experience of time slows. If you could accelerate a massive particle to the speed of light, its mass would become infinite, and time would stop for the particle. “If” is the key word there: it can’t. It would require an infinite amount of energy to accelerate it to the speed of light.

But light has no mass. Relativity describes a strange property of the universe, which is hard to wrap your head around. Light always moves at the same speed, no matter your perspective. Take two spacecraft in outer space, which are completely stationary relative to each other. Shine a laser from one, and measure how long it takes for the light to get to the other. How fast is it going? Roughly 186,000 miles/second. Now, start one ship moving away from the other at half the speed of light. Repeat the experiment. One ship is moving away from the other at a speed of 93,000 miles/second. From the perspective of the moving ship, how fast is the light moving away from it towards the other ship? 186,000 miles/second. From the perspective of the stationary ship, how fast is the laser light approaching it? 186,000 miles/second.

It’s not that there’s some magic thing about light that makes it move while time stops for it. Light is massless, so it can move at the speed of light. Time dilation doesn’t apply because it has no mass.

But even if that weren’t the case, that’s got nothing to do with dimensionality. Dimensionality is a direction: what does this rubbish have to do with the different directions that light can move in? Absolutely nothing: the way he’s using the word “dimension” has nothing to do with what dimensions mean.

All “objects” or instances of matter are time-bound; they change, or die, or dissolve, or evaporate. Because they are subject to time, objects can be called four-dimensional.

Nope.

Everything in our universe is subject to time, because time is one of the dimensions in our universe. Time is a direction that we move. We don’t have direct control over it – but it’s still a direction. When and where did I write this blog post compared to where I am when you’re reading it? The only way you can specify that is by saying how far my position has changed in four directions: 3 spatial directions, and time. Time is a dimension, and everything in our universe needs to consider it, because you can’t specify anything in our universe without all four dimensions.

The enormous energy that can be released from a tiny object (as in an atomic bomb) demonstrates the role dimensions play in constructing reality.

No: the enormous energy that can be released from a tiny object demonstrates the fact that a small quantity of matter is equivalent to a large quantity of energy. As you’d expect if you look at that original equation: e=mc^2. A gram of mass – something the size of a paperclip – is equivalent to about 25 million kilowatt-hours of energy – or more than the total yearly energy use of 1,200 average americans. That’s damned impressive and profound, without needing to draw in any mangled notions of dimensions or magical dimensional powers.

Higher dimensions are mind-blowingly powerful; even infinitely so. Such power is so expansive that it can’t have form, definition, or identity, like a ball of uranium or a human being, without finding expression in lower dimensions. The limitations of time and space allow infinite power to do something other than constantly annihilate itself.

Do I even need to respond to this?

Einstein’s equation E=mc2 bridges the fourth and the fifth dimensions, expressed as matter and energy. Imagine a discovery that bridges expressions of the fifth and sixth dimensions, such as energy and consciousness. Consciousness has the five-dimensional qualities of energy, but it can’t be “spent” in the way energy can because it doesn’t change form the way energy does. Therefore, it’s limitless.

And now we move from crackpottery to mysticism. Einstein’s mass-energy equation doesn’t bridge dimensions, and dimensionality has nothing do with mass-energy equivalence. And now our crackpot friend suddenly throws in another claim, that consciousness is the sixth dimension? Or consciousness is the bridge between the fifth and sixth dimensions? It’s hard to figure out just what he’s saying here, except for the fact that it’s got nothing to do with actual dimensions.

Is there a sixth dimension? Who knows? According to some modern theories, our universe actually has many more than the 4 dimensions that we directly experience. There could be 6 or 10 or 20 dimensions. But if there are, those dimensions are just other directions that things can move. They’re not abstract concepts like “consciousness”.

And of course, this is also remarkably sloppy logic:

  1. Consciousness has the 5-dimensional qualities of energy
  2. Consciousness can’t be spent.
  3. Consciousness can’t change form.
  4. Therefore consciousness is unlimited.

The first three statements are just blind assertions, given without evidence or argument. The fourth is presented as a conclusion drawn from the first three – but it’s a non-sequitur. There’s no real way to conclude the last statement given the first three. Even if you give him all the rope in the world, and accept those three statements as axioms – it’s still garbage.

The Intellectual Gravity of Brilliant Baseball Players

Some of my friends at work are baseball fans. I totally don’t get baseball – to me, it’s about as interesting as watching paint dry. But thankfully, some of my friends disagree, which is how I found this lovely little bit of crackpottery.

You see, there’s a (former?) baseball player named Jose Canseco, who’s been plastering twitter with his deep thoughts about science.

I may not be 100% right but think about it.How else could 30 foot leather birds fly?

— Jose Canseco (@JoseCanseco) February 19, 2013

At first glance, this is funny, but not particularly interesting. I mean, it's a classic example of my mantra: the worst math is no math.

The core of this argument is pseudo-mathematical. The dumbass wants to make the argument that under current gravity, it wouldn't be possible for things the size of the dinosaurs to move around. The problem with this argument is that there's no problem! Things the size of dinosaurs could move about in current gravity with absolutely no difficult. If you actually do the math, it's fine.

If dinosaurs had the anatomy of human beings, then it's true that if you scaled them up, they wouldn't be able to walk. But they didn't. They had anatomical structures that were quite different from ours in order to support their massive size. For example, here's a bone from quetzlcoatlus:

Media,111639,en See the massive knob sticking out to the left? That's a muscle attachement point. That gave the muscles much greater torque than ours have, which they needed. (Yes, I know that Quetzalcoatlus wwasn't really a dinosaur, but it is one of the kinds of animals that Canseco was talking about, and it was easy to find a really clear image.)

Most animal joints are, essentially, lever systems. Muscles attach to two different bones, which are connected by a hinge. The muscle attachement points stick out relative to the joint. When the muscles contract, that creates a torque which rotate the bones around the joint.

The lever is one of the most fundamental machines in the universe. It operates by the principal of torque. Our regular daily experiences show that levers act in a way that magnifies our efforts. I can't walk up to a car and lift it. But with a lever, I can. Muscle attachment points are levers. Take another look at that bone picture: what you're seeing is a massive level to magnify the efforts of the muscles. That's all that a large animal needed to be able to move around in earths gravity.

This isn't just speculation - this is stuff that's been modeled in great detail. And it's stuff that can be observed in modern day animals. Look at the skeleton of an elephant, and compare it to the skeleton of a dog. The gross structure is very similar - they are both quadripedal mammals. But if you look at the bones, the muscle attachment points in the elephants skeleton have much larger projections, to give the muscles greater torque. Likewise, compare the skeleton of an american robin with the skeleton of a mute swan: the swan (which has a maximum recorded wingspan of 8 feet!) has much larger projections on the attachment points for its muscles. If you just scaled a robin from its 12 inch wingspan to the 8 feet wingspan of a swan, it wouldn't be able to walk, much less fly! But the larger bird's anatomy is different in order to support its size - and it can and does fly with those 8 foot wings!

That means that on the basic argument for needing different gravity, Canseco fails miserably.

Canseco's argument for how gravity allegedly changed is even worse.

What he claims is that at the time when the continental land masses were joined together as the pangea supercontinent, the earths core moved to counterbalance the weight of the continents. Since the earths core was, after this shift, farther from the surface, the gravity at the surface would be smaller.

This is an amusingly ridiculous idea. It's even worse that Ted Holden and his reduced-felt-gravity because of the electromagnetic green saturn-star.

First, the earths core isn't some lump of stuff that can putter around. The earth is a solid ball of material. It's not like a ball of powdered chalk with a solid lump of uranium at the center. The core can't move.

Even if it could, Canseco is wrong. Canseco is playing with two different schemes of how gravity works. We can approximate the behavior of gravity on earth by assuming that the earth is a point: for most purposes, gravity behaves almost as if the entire mass of the earth was concentrated at the earths center of mass. Canseco is using this idea when he moves the "core" further from the surface. He's using the idea that the core (which surrounds the center of mass in the real world) is the center of mass. So if the core moves, and the center of mass moves with it, then the point-approximation of gravity will change because the distance from the center of mass has increased.

But: the reason that he claims the core moved is because it was responding to the combined landmasses on the surface clumping together as pangea. That argument is based on the idea that the core had to move to balance the continents. In that case, the center of gravity wouldn't be any different - if the core could move to counterbalance the continents, it would move just enough to keep the center of gravity where it was - so if you were using the point approximation of gravity, it would be unaffected by the shift.

He's combining incompatible assumptions. To justify moving the earths core, he's *not* using a point-model of gravity. He's assuming that the mass of the earths core and the mass of the continents are different. When he wants to talk about the effect of gravity of an animal on the surface, he wants to treat the full mass of the earth as a point source - and he wants that point source to be located at the core.

It doesn't work that way.

The thing that I find most interesting about this particular bit of crackpottery isn't really about this particular bit of crackpottery, but about the family of crackpottery that it belongs to.

People are fascinated by the giant creatures that used to live on the earth. Intuitively, because we don't see giant animals in the world around us, there's a natural tendency to ask "Why?". And being the pattern-seekers that we are, we intuitively believe that there must be a reason why the animals back then were huge, but the animals today aren't. It can't just be random chance. So people keep coming up with reasons. Like:

  1. Neal Adams: who argues that the earth is constantly growing larger, and that gravity is an illusion caused by that growth. One of the reasons, according to his "theory", for why we know that gravity is just an illusion, is because the dinosaurs supposedly couldn't walk in current gravity.
  2. Ted Holden and the Neo-Velikovskians: who argue that the solar system is drastically different today than it used to be. According to Holden, Saturn used to be a "hyperintelligent green electromagnetic start", and the earth used to be tide-locked in orbit around it. As a result, the felt effect of gravity was weaker.
  3. Stephen Hurrell, who argues similarly to Neal Adams that the earth is growing. Hurrell doesn't dispute the existence of gravity the way that Adams does, but similarly argues that dinosaurs couldn't walk in present day gravity, and resorts to an expanding earth to explain how gravity could have been weaker.
  4. Ramin Amir Mardfar: who claims that the earth's mass has been continually increasing because meteors add mass to the earth.
  5. Gunther Bildmeyer, who argues that gravity is really an electromagnetic effect, and so the known fluctuations in the earths magnetic fields change gravity. According to him, the dinosaurs could only exist because of the state of the magnetic field at the time, which reduced gravity.

There are many others. All of them grasping at straws, trying to explain something that doesn't need explaining, if only they'd bother to do the damned math, and see that all it takes is a relatively small anatomical change.

Euler's Equation Crackpottery

One of my twitter followers sent me an interesting piece of crackpottery. I debated whether to do anything with it. The thing about crackpottery is that it really needs to have some content. Total incoherence isn’t amusing. This bit is, frankly, right on the line.

Euler’s Equation and the Reality of Nature.

a) Euler’s Equation as a mathematical reality.

Euler’s identity is “the gold standard for mathematical beauty’.
Euler’s identity is “the most famous formula in all mathematics”.
‘ . . . this equation is the mathematical analogue of Leonardo
da Vinci’s Mona Lisa painting or Michelangelo’s statue of David’
‘It is God’s equation’, ‘our jewel ‘, ‘ It is a mathematical icon’.
. . . . etc.

b) Euler’s Equation as a physical reality.

“it is absolutely paradoxical; we cannot understand it,
and we don’t know what it means, . . . . .’
‘ Euler’s Equation reaches down into the very depths of existence’
‘ Is Euler’s Equation about fundamental matters?’
‘It would be nice to understand Euler’s Identity as a physical process
using physics.‘
‘ Is it possible to unite Euler’s Identity with physics, quantum physics ?’

My aim is to understand the reality of nature.

Can Euler’s equation explain me something about reality?

To give the answer to this. question I need to bind Euler’s equation with an object – particle. Can it be math- point or string- particle or triangle-particle? No, Euler’s formula has quantity (pi) which says me that the particle must be only a circle .

Now I want to understand the behavior of circle – particle and therefore I need to use spatial relativity and quantum theories. These two theories say me that the reason of circle – particle’s movement is its own inner impulse (h) or (h*=h/2pi).

a) Using its own inner impulse (h) circle – particle moves ( as a wheel) in a straight line with constant speed c = 1. We call such particle – ‘photon’. From Earth – gravity point of view this speed is maximally. From Vacuum point of view this speed is minimally. In this movement quantum of light behave as a corpuscular (no charge).

b) Using its own inner impulse / intrinsic angular momentum ( h* = h / 2pi ) circle – particle rotates around its axis. In such movement particle has charge, produce electric waves ( waves property of particle) and its speed ( frequency) is : c.

1. We call such particle – ‘ electron’ and its energy is: E=h*f.

In this way I can understand the reality of nature.

==.

Best wishes.

Israel Sadovnik Socratus.

Euler’s equation says that e^{ipi} + 1 = 0. It’s an amazingly profound equation. The way that it draws together fundamental concepts is beautiful and surprising.

But it’s not nearly as mysterious as our loonie-toon makes it out to be. The natural logarithm-base is deeply embedded in the structure of numbers, and we’ve known that, and we’ve known how it works for a long time. What Euler did was show the relationship between e and the fundamental rotation group of the complex numbers. There are a couple of ways of restating the definition of that make the meaning of that relationship clearer.

For example:

e^z = lim_{nrightarrow infty}(1 + frac{z}{n})^n

That’s an alternative definition of what e is. If we use that, and we plug ipi into it, we get:

e^{ipi} = lim_{n rightarrow infty}(1+frac{ipi}{n})^n

If you work out that limit, it’s -1. Also, if you take values of N, and plot (1 + frac{ipi}{n})^1, (1+frac{ipi}{n})^2, (1 + frac{ipi}{n})^3, and (1 + frac{ipi}{n})^4, … on the complex plane, as N gets larger, the resulting curve gets closer and closer to a semicircle.

An equivalent way of seeing it is that exponents of e^i are rotations in the complex number plane. The reason that e^{ipi} = -1 is because if you take the complex number (1 + 0i), and rotate it by pi radians, you get -1: 1(e^{ipi}) = -1.

That’s what Euler’s equation means. It’s amazing and beautiful, but it’s not all that difficult to understand. It’s not mysterious in the sense that our crackpot friend thinks it is.

But what really sets me off is the idea that it must have some meaning in physics. That’s silly. It doesn’t matter what the physical laws of the universe are: the values of pi and e will not change. It’s like trying to say that there must be something special about our universe that makes 1 + 1 = 2 – or, conversely, that the fact that 1+1=2 means something special about the universe we live in. These things are facts of numbers, which are independent of physical reality. Create a universe with different values for all of the fundamental constants – e and π will be exactly the same. Create a universe with less matter – e and π will still be the same. Create a universe with no matter, a universe with different kinds of matter, a universe with 300 forces instead of the four that we see – and e and π won’t change.

What things like e and π, and their relationship via Euler’s equation tell us is that there’s a fundamental relationship between numbers and shapes on a two-dimensional plane which does not and cannot really exist in the world we live in.

Beyond that, what he’s saying is utter rubbish. For example:

These two theories say me that the reason of circle – particle’s movement is its own inner impulse (h) or (h*=h/2pi). Using its own inner impulse (h) circle – particle moves ( as a wheel) in a straight line with constant speed c = 1. We call such particle – ‘photon’. From Earth – gravity point of view this speed is maximally. From Vacuum point of view this speed is minimally. In this movement quantum of light behave as a corpuscular (no charge).

This is utterly meaningless. It’s a jumble of words that pretends to be meaningful and mathematical, when in fact it’s just a string of syllables strung together nonsensical ways.

There’s a lot that we know about how photons behave. There’s also a lot that we don’t know about photons. This word salad tells us exactly nothing about photons. In the classic phrase, it’s not even wrong: what it says doesn’t have enough meaning to be wrong. What is the “inner impulse” of a photon according to this crackpot? We can’t know: the term isn’t defined. We are pretty certain that a photon is not a wheel rolling along. Is that what the crank is saying? We can’t be sure. And that’s the problem with this kind of crankery.

As I always say: the very worst math is no math. This is a perfect example. He starts with a beautiful mathematical fact. He uses it to jump to a completely non-mathematical conclusion. But he writes a couple of mathematical symbols, to pretend that he’s using math.

The Meta of Gödel

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:

  1. Take a simple logic. We’ve been using a variant of the Principia Mathematica’s logic, because that’s what Gödel used.
  2. 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.
  3. 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.
  4. 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, xn, 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:

  1. Zero: ¬(succ(x1) = 0)
  2. Uniqueness: succ(x1) = succ(y1) Rightarrow x = y
  3. Induction: x2(0) ∧ ∀x1(x2(x1)⇒ x2(succ(x1))) ⇒ ∀x1(x2(x1))
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:

  1. p lor p Rightarrow p
  2. p Rightarrow p lor q
  3. p lor q Rightarrow p lor q
  4. (p Rightarrow q) Rightarrow (r lor p Rightarrow r lor q)

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:

  1. forall v(a) Rightarrow text{subst}[v/c](a)
  2. forall v(b lor a) Rightarrow (b or forall v(a))
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 a, exists u (forall v (u(v) Leftrightarrow a. 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 forall x_1 (x_2(x_1) Leftrightarrow y_2(y_1) Rightarrow x_2 = x_1). 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.

Defining Properties Arithmetically (part 1): Gödel and Primitive Recursion

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 phi(x_1, x_2, ..., x_n) is defined via primitive recursion if, for some other formulae psi and mu:

  • Base: phi(0, x_2, ..., x_n) = psi(x_2, ..., x_n)
  • Recursive: phi(i+1, x_2, ..., x_n) = mu(i, phi(i, x_2, ..., x_n), x_2, ..., x_n).

So, basically, the first parameter is a bound on the number of times that phi can invoked recursively. When it’s 0, you can’t invoke phi any more.

A formula is primitive recursive if it defined from a collection of formulae phi_1, ..., phi_n where any formula phi_i is defined via primitive recursion from phi_1, ..., phi_{i-1}, or the primitive succ function from Peano arithmetic.

For any formula phi_i 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: R(x_1, ..., x_n) is primitive recursive if and only if there exists a primitive recursive function phi such that phi(x_1, ..., x_n) = 0.

With primitive recursive formulae and relations defined, there’s a bunch of theorems about how you can compose primitive recursive formulae and relations:

  1. 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.
  2. If R and S are primitive relations, then ¬R, R∧S, R∨S are all primitive recursive.
  3. If phi(x_1, ..., x_n) and psi(x_1, ..., x_n) are primitive recursive functions, then the relation R(x_1, ..., x_n) Leftrightarrow (phi(x_1, ..., x_n) = psi(x_1, ..., x_n) is also primitive recursive.
  4. Let xv and zv be finite-length tuples of variables. If the function phi(xv) and the relation R(y, zv) are primitive recursive, then so are the relations:
    • S(xv, zv) Leftrightarrow  (exists y le phi(xv). R(y, zv))
    • T(xv, zv) Leftrightarrow  (forall y le A(xv). R(y, zv))
  5. Let xv and zv be finite-length tuples of variables. And let text{argmin}[y le f(x).R(x)] be the smallest value of x for which y le f(x) and R(x) is true, or 0 if there is no such value. Then if the function phi(xv) and the relation R(y, zv) are primitive recursive, then so is the function P(xv, zv) = (text{argmin}[y le A(xv). R(y, zv))].

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!

G&oum;del Numbering: Encoding Logic as Numbers

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 a1, b1.
  • 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 a2, b2, …
  • 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(a1), 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 → signn+1(signn)
 formula → ¬(elementary_formula)
 formula → (elementary_formula) or (elementary_formula)
 formula → ∀ signn (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.
    1. ¬(succ(x1) = 0): 0 is a natural number, and it’s not the successor of anything.
    2. succ(x1) = succ(y1) ⇒ x1 = y1: Successors are unique.
    3. (x2(0) ∧ ∀ x1 (x2(x1) ⇒ x2(succ(x1)))) ⇒ ∀ x1(x2(x1)): 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.
    <ol>
       <li> <em>p &or; p &rArr; p</em></li>
       <li> <em>p &rArr; p &or; q</em></li>
       <li> <em>p &or; q &rArr; q &or; p</em></li>
       <li> <em>(p &rArr; q) &rArr; (p &or; r) &rArr; (q &or; r)</em></li>
    
  • 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.
    1. ∀ 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.
    2. (∀ 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>&forall; x<sub>i</sub> (x<sub>i+1</sub>(x<sub>i</sub>) &hArr; y<sub>i+1</sub>(y<sub>i</sub>)) &rArr; 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, p2 will represent a type two variable, p3 will represent a type-3 variable, etc.

    Using those symbol encodings, we can take a formula written as symbols x1x2x3…xn, and encode it numerically as the product 2x13x25x2…pnxn, where pn is the nth prime number.

    For example, suppose I wanted to encode the formula: ∀ x1 (y2(x1)) ∨ x2(x1).

    First, I’d need to encode each symbol:

    1. “∀” would be 9.
    2. “x1“” = 17
    3. “(” = 11
    4. “y2” = 192 = 361
    5. “(” = 11
    6. “x1” = 17
    7. “)” = 13
    8. “∨” = 7
    9. “x2” = 172 = 289
    10. “(” = 11
    11. “x1” = 17
    12. “)” = 13
    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 29 317 511 7361 1111 1317 1713 197 23289 2911 3117 3713 4113, which according to Hugs is the number (warning: you need to scroll to see it. a lot!):

    1,821,987,637,902,623,701,225,904,240,019,813,969,080,617,900,348,538,321,073,935,587,788,506,071,830,879,280,904,480,021,357,988,798,547,730,537,019,170,876,649,747,729,076,171,560,080,529,593,160,658,600,674,198,729,426,618,685,737,248,773,404,008,081,519,218,775,692,945,684,706,455,129,294,628,924,575,925,909,585,830,321,614,047,772,585,327,805,405,377,809,182,961,310,697,978,238,941,231,290,173,227,390,750,547,696,657,645,077,277,386,815,869,624,389,931,352,799,230,949,892,054,634,638,136,137,995,254,523,486,502,753,268,687,845,320,296,600,343,871,556,546,425,114,643,587,820,633,133,232,999,109,544,763,520,253,057,252,248,982,267,078,202,089,525,667,161,691,850,572,084,153,306,622,226,987,931,223,193,578,450,852,038,578,983,945,920,534,096,055,419,823,281,786,399,855,459,394,948,921,598,228,615,703,317,657,117,593,084,977,371,635,801,831,244,944,526,230,994,115,900,720,026,901,352,169,637,434,441,791,307,175,579,916,097,240,141,893,510,281,613,711,253,660,054,258,685,889,469,896,461,087,297,563,191,813,037,946,176,250,108,137,458,848,099,487,488,503,799,293,003,562,875,320,575,790,915,778,093,569,309,011,025,000,000,000.

    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.

    Gödel's Incompleteness

    I’ve mentioned Gödel’s incompleteness theorems many times on this blog, but I’ve never actually written about them in detail. I was asking, on twitter, for topics that readers would be interested in, and one thing that came up is actually go through the proof of incompleteness. I’m happy to take the time to do that incompleteness is one of the most beautiful and profound proofs that I’ve ever seen.

    It takes a fair bit of effort though, so it’s going to take a couple of posts just to get through the preliminaries. 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.

    The story of incompleteness is really pretty interesting. In the late 19th and early 20th centuries, mathematicians started to get really interested in formal foundations. Why, exactly, this happened is somewhat of a debate, but the story I find most likely is that it’s a result of Cantor and his naive set theory.

    Set theory came along, and it seemed like such a great idea. It seems incredibly simple, and it makes many proofs in many different subject areas appear to be really simple. But there’s a huge problem: it’s not consistent.

    The problem is what’s come to be known as Russell’s paradox. I’ve explained Russell’s paradox a bunch of times on the blog. For this post, I’m going to do it a bit differently. As part of my prep for this series of posts, I’ve been reading the book Gödel’s Proof, and in that book, they have a way of rephrasing Russell’s paradox which I think clarifies it nicely, so I’m going to try their version.

    If you think about the set of all sets, you can partition it into two non-overlapping subsets. There’s a subset of normal sets, and a class of abnormal sets. What makes a set abnormal is that it’s self-referential – an abnormal set contains itself as a member.

    So considered the set of all normal sets. Is it, itself, a normal set?

    The answer is: argh! Because if the set of all normal sets is normal, then it must be a member of itself. But if it’s a member of itself, it can’t be normal. Similarly, if you start off saying that the set of all normal sets isn’t normal, then by definition, it is normal. No matter what you do, you’re screwed.

    What 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 Cantor incident.

    So, eventually, along came two mathematicians, Russell and Whitehead, and they created an amazing piece of work called the Principia Mathematica. The principia was supposed to be an ideal, perfect mathematical foundation. The Principia was supposed to have two 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. You 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 normal sets – because that set is formed using a predicate – and when you’re forming a set like “the set of proper sets”, 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.

    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.

    1. 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.
    2. 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!
    3. 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.

    We’re going to go through all of that in detail. But to do that in a way where we can really understand it, we’re going to need to start by looking at the logic used by the Principia. Then we’ll look at how to encode Principia-logical statements as natural numbers using a technique called Gödel numbering, and how to express predicates numerically. And then, finally, we can look at how you can form the classic Gödel statement that shows that the Principia’s system of logic is incomplete.