What is it about natural numbers that makes them a prerequisite for Godel's Incompleteness Theorem to hold?

Question

I came across an old question on here asking about Gödel's Incompleteness Theorem and the theory of Real Closed Fields, specifically about why the former doesn't apply to the latter. The top answer explained that the theory of Real Closed Fields avoids the Incompleteness Theorem because it lacks the ability to distinguish between natural numbers and non-natural numbers.

I understand that this inability to pick out a distinct class of natural numbers is relevant because the Incompleteness Theorem requires a formal system to contain natural number arithmetic of a certain complexity, but I'm curious about why the theorem specifically requires natural numbers. I have a very basic understanding of the mechanics behind the Incompleteness Theorem: I know that Gödel's strategy was to encode formulas by mapping symbols to numbers and then using these numbers as the exponents in a product of prime numbers. However, I'm not sure what specific property of natural numbers is required here, beyond some vague intuition about discreteness and the sense that the theory of Real Closed Fields may be "too analog" for Gödel encoding to work.

Is this intuition on the right track? What is it about the distinction between real & natural numbers that determines whether or not the Incompleteness Theorems apply?

Edit: After submitting this question, it occurred to me that perhaps the answer has something to do with the way Gödel encoding relies on the unique factorization theorem to ensure a unique mapping for every sequence of characters. Is this why natural numbers are required?

Answer 1

Fundamentally what is needed for the incompleteness theorem argument to go through is that it is possible to use arithmetic to talk about Turing machines. This is explained e.g. by Scott Aaronson:

Alright, should I let you in on a secret? The proof of the Incompleteness Theorem is about two lines. The caveat is that, to give the two-line proof, you first need the concept of a computer.

In particular, Peano arithmetic is capable of making statements of the form "this Turing machine halts on this input" or "this Turing machine halts on some input," etc. Consider a Turing machine $T$ which takes as input another Turing machine and searches for a proof in Peano arithmetic that that Turing machine will halt or not (this is possible because the axioms of Peano arithmetic themselves are computable, so Turing machines can discuss them). Now we can ask: does $T$ halt on every input or run forever on some input?

• If $T$ always halts, then (assuming Peano arithmetic is sound!) $T$ can solve the halting problem. But we know that that's impossible. On the other hand:
• If $T$ runs forever on some Turing machine $S$, then Peano arithmetic can't either prove or disprove the statement "$S$ halts," so we've found a statement independent of PA.

This is the first incompleteness theorem. You can click the link to see Scott Aaronson explain the proof of the second incompleteness theorem using a slightly different Turing machine $T'$.

This argument is extremely general. It applies to any consistent formal system which is 1) computable, so Turing machines can talk about it, and 2) powerful enough to talk about Turing machines. These are the "recursive" and "sufficiently powerful" hypotheses in the incompleteness theorem, and as this argument makes clear they are both essential. It is extremely common to see informal statements of the incompleteness theorem which drop one or both of these hypotheses!

So, now the question is: what is it about Peano arithmetic that makes it possible to talk about Turing machines in it, while the same is not true of the theory of real closed fields (RCF)? There are different ways to do this but they all start with some variation of the fact that you can use a natural number to describe a finite list of natural numbers. Godel did this using prime factorizations but this is inessential; any bijection from natural numbers to finite lists of natural numbers that can be described in Peano arithmetic can be used here, e.g. interleaving binary digits.

RCF is much too weak to do anything like this (because its theory is decidable by Tarski's theorem); you might think that we have even more ability to do this here because a real number has infinitely many digits so could store an infinite amount of information, but we can't access any of this information in RCF. In fact it is not possible to talk about the digits of a real number in RCF; RCF can basically only talk about polynomials and the digits, as functions of a real number, are not continuous so are not polynomials.