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I read in another post (Why don't we use Presburger's arithmetic instead of Peano's arithmetic?) the following exposition:

Godel's incompleteness theorem, philosophically (and morally) speaking it says that a consistent theory cannot have all the following properties:

  • Can be handled algorithmically (not too complicated)

  • Interesting as a foundational theory (can prove useful statements)

  • Complete (every statement over it is provable or disprovable)

I am wondering if there is any notable extensions of PA that satisfies the last two properties mentioned above? Alternatively, is there a different set of axioms that satisfies the last two properties mentioned above?

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J.Qu
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1 Answers1

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The most interesting complete extension of PA is of course the complete theory of the natural numbers: $$\text{Th}(\mathbb{N}) = \{\varphi\text{ a sentence in the language of arithmetic}\mid \varphi \text{ is true in }\mathbb{N}\}$$

This theory is complete because for any sentence $\varphi$, either $\varphi$ or $\lnot \varphi$ is true in $\mathbb{N}$, so $\varphi\in \text{Th}(\mathbb{N})$ or $\lnot\varphi \in \text{Th}(\mathbb{N})$. And it's certainly interesting: it can prove everything PA can and more - in fact, it answers all questions of number theory that can be expressed as first-order sentences in the language of arithmetic!

The deficiency of this theory as a foundation for mathematics is that we have no way to understand what the axioms are, i.e. whether $\varphi$ or $\lnot \varphi$ is an axiom for each sentence $\varphi$. That is to say, it fails the first property on your list: it's too complicated.

Alex Kruckman
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