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For example, we define 'what a set is' or 'what natural numbers are' using a bunch of assumptions about how they should behave. Those assumptions are all there is to the 'world or sets' or 'world of natural numbers'. So we cannot assert that a certain behavior of 'sets' or 'natural numbers' is true if it does not follow from our assumptions. The Incompleteness theorem seems really unremarkable in this sense. Mathematical structures are thought up by humans. Suppose we think of a universe where movement is not possible and we set up the rules for the behavior of the member particles of the universe. Then Incompleteness theorem is like saying "the thorey does not describe movement behavior of particles", even where there is no such thing as a 'movement behavior' in our thought-up world.

What am I missing something? Why is this theorem remarkable?

Andrés E. Caicedo
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Ryder Rude
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  • The incompleteness theorem is not concerned with statements that fall outside the relevant axiom system. Rather it concerns statements which are True within the axioms (so that no counterexample can be constructed within the axiom system) but which are not provable if you stick to the axioms. – lulu Mar 17 '20 at 15:26
  • @lulu but how is 'truth' defined independent of provability from the axioms? Axioms define how the world of sets will behave. 'Provability from axioms' defines what is true and what is not – Ryder Rude Mar 17 '20 at 15:27
  • True meaning that the negation can not be proven within the relevant axioms. True is not synonymous with "provable", that is precisely the point. – lulu Mar 17 '20 at 15:29
  • As far as I am aware, for instance, it is perfectly possible that it is true that there are infinitely many twin prime pairs but that the axioms of arithmetic are insufficient to prove it. I don't believe that personally...I think better tools will resolve the matter. But I don't know more than anyone else. – lulu Mar 17 '20 at 15:36
  • @lulu Then that should that mean that the axioms of arithmetic define a world of natural numbers which doesn't say anything about whether there are infinitely many twin primes or not. You could extend this world using the same axioms to define another world where there are not infinitely many twin primes. Or you could define a world of natural numbers with the same axioms but with infinitely many twin primes. The statement 'there are infinitely many twin primes' is true only under the latter definition of natural numbers. – Ryder Rude Mar 17 '20 at 15:45
  • Not following. Yes, there might be other axiom systems in which there are finitely or infinitely many twin primes. So what? We are interested in the axiom system we use regularly. – lulu Mar 17 '20 at 15:53
  • "True" means that no counterexamples exist. "Provable" means it can be demonstrated from the axioms. Entirely different concepts. – saulspatz Mar 17 '20 at 16:42
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    I disagree with the second sentence of the question. Assumptions are not all there is to the world of sets or the world of natural numbers. The important ingredients in these worlds are the sets and the natural numbers, respectively. Assumptions are our attempts to summarize some of the facts about these worlds, so that we can reason from those facts to further (usually more complicated) information. – Andreas Blass Mar 17 '20 at 17:04

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Lots of questions very similar to this one have been asked on this site, but at the moment I can't find an exact duplicate. So here goes:

I think the key observation (I'll make another one below) is the following:

We can dodge the philosophical issues here entirely.

One can dodge the entire philosophical discussion by noting that there is a "purely formalist" version of Godel's incompleteness theorem, which makes no reference to truth at all. Namely:

(Godel, + Rosser/Robinson) There is no consistent computably axiomatizable complete theory containing $Q$.

More snappily, $Q$ is essentially incomplete. (This can actually be improved substantially - $Q$ can be replaced by even weaker systems, and "containing" can be replaced by "interpreting" - but let's ignore this for now.)

This is still fairly unsurprising to a modern audience, who are familiar with computability theory and recognize that this is more-or-less the incomputability of the halting problem in disguise, but in $1931$ that notion wasn't yet well-developed. Without it, essential incompleteness is in my opinion quite counterintuitive - why shouldn't there be a "simplified" arithmetic system (regardless of whether it matches our intuitions about the natural numbers, or - adopting a Platonist standpoint - whether it is true of $\mathbb{N}$)?

OK, now let me push back against the philosophical assumptions in the question itself. As noted above, this isn't necessary to defend the meaningfulness and interestingness of incompleteness, but it's still worth doing. The point is:

Not everyone agrees with you here.

This question adopts a formalist stance ("Those assumptions are all there is to the 'world or sets' or 'world of natural numbers'"). This is not universal: there are for example Platonist, or "Platonist-for-$\mathbb{N}$," mathematicians out there, regardless of how strange that may strike you. To these mathematicians your dismissal of incompleteness is largely beside the point.

Now you might respond that you don't care since Platonism is "obviously silly" - there are also plenty of mathematicians who take this stance - but I don't think that gets you out of this entirely. Even from the standpoint of formalism we can still argue that one crucial feature of mathematics is its universality despite philosophical differences (a Platonist and formalist can work together to prove the same theorems). From this perspective the fact that the incompleteness theorem is compelling from at least one philosophical stance (Platonism) means that it can't be rejected as uninteresting even from a radically different stance (formalism). Again, this may seem weird but there are serious mathematicians who hold it (at least one, to be precise :P).

But, as noted above, we don't even need this.

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Noah Schweber
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