I've been getting extremely conflicting answers about this. Please help me. A summary of the different viewpoints:

  1. It seems like most people are taking the Platonic viewpoint : that natural numbers are taken for granted and statements about natural numbers are true or false in the absolute sense. They are false if a counter-example exists. And that we do not require any axiomatic system to make precise the notions of "natural numbers", "there exists", "counter-example", etc. All of it comes from human intuition about natural numbers

  2. The second viewpiont is we really have two axiomatic systems: A strong one like $ZFC$ and a weaker one like Peano arithmetic. The stronger model is taken to define natural numbers. Anything that's provable about the naturals in the stronger model is referred to as the "truths" about naturals. So the weaker axiomatic system like Peano is incomplete in the sense that it cannot derive all the truths of the stronger model. Also there is the non-standard model of arithmetic. So, the reason Peano axioms are incomplete is really because there are no absolute truths about natural numbers. The truths depend on our definition of natural numbers according to the stronger theory. Does the non-standard model of arithmetic disprove the Platonic viewpoint?

What I find weird about the second viewpoint is that it makes Godel's incompleteness theorem seem trivial. It means the thoeorem just says that certain statements about natural numbers may be independent from Peano arithmetic in the same sense the continuum hypothesis is independent from ZFC. Those statements are true in some models and false in others.

If the second viewpoint is correct, then does it mean that human intuition about natural numbers is actually ambiguous? Humans feel like we just know, without any reference to any axiomatic system, what it means for statements about natural numbers to be true or false : it's just false if you can find a counter-example. Is this intuition wrong and the statements are actually ambgious (due to e.g. the non standard model of arithmetic)?

OR Is the intuition actually correct and it's just that any axiomatic system (even ZFC) is insufficient to model the the natural numbers that unambiguously exist in human intuition. If yes, on what basis can we claim this?

And since Godel's incompleteness theorem is related to the halting problem, how do the two viewpoints relate to the halting problem? It seems like the halting problem is making an absolute statement about the limits of computability using Turing machines (e.g. the Busy Beaver function is non-computable in the absolute sense).

  • $\begingroup$ "since Godel's incompleteness theorem is equivalent to the halting problem" What does that even mean? $\endgroup$ Feb 25 at 5:16
  • $\begingroup$ @NoahSchweber I've changed it to "related". Most people say they're related. $\endgroup$
    – Ryder Rude
    Feb 25 at 5:17

1 Answer 1


When there are multiple perspectives on mathematics that are compatible in the sense that mathematicians with these perspectives can still collaborate, you're free to choose the perspective that makes most sense to you. Neither is right or wrong.

Saying that a statement of the form $\forall x: \phi(x)$ is false iff you can find a counterexample is not very practical, though. What if there is a counterexample, but it is too big for you to write down? This is not too different from there existing a counterexample, but it being nonstandard.

We have strong intuitions about small natural numbers, but it's on the large (infinite) end where different models of e.g. Peano Arithmetic start to differ.

This issue also shows up with Turing machines: how can we tell the difference between a machine that takes an unwieldy amount of time to halt, and one that doesn't halt at all? Nonstandard models of arithmetic have Turing machines that take nonstandard amounts of time $t$ to halt: so to smaller models that don't have the number $t$ it looks like those Turing machines don't halt.


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