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 :
- 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.