4

Godel's first incompleteness theorem states roughly that you can't write down a finite list of axioms that can decide all statements about arithmetic: any such formal system is incomplete. I feel like I have a pretty good understanding of what this means concretely, but that's because there are concrete examples of formal systems that are complete.

Godel's second incompleteness theorem states that no formal system that can talk about arithmetic can prove its own consistency. I don't really appreciate this one as much on a gut level because I sort of feel like, well, duh - how could any formal system possibly prove its own consistency? Of course I know about Godel numbering and what not, but still, what I'd really like is an explicit example of a simple formal system, with a proof in that system of the system's own consistency. Does such a thing exist?

Jack M
  • 27,819
  • 7
  • 63
  • 129
  • See wikipedia on "self-verifying theories." Dan Willard has done a lot of research on this, and on trying to figure out sharp limits on how powerful a theory can be before running into the second incompleteness theorem. These theories have just enough arithmetic to talk about Godel numbers and provability internally without being able to perform the diagonalization. These theories are to an extent "not natural" and are specifically made for this purpose. – penguin_surprise Sep 14 '18 at 08:26
  • 5
    An inconsistent system will easily prove its own consistency (as well as everything else), so take PA and add the axiom $1=0$. – hmakholm left over Monica Sep 14 '18 at 08:37
  • Presburg-Arithmetic is known to be consistent, but I do not know whether it can prove its own consistency. – Peter Sep 14 '18 at 08:38
  • @HenningMakholm Let's say we're looking for systems with consistency proofs which we're inclined philosophically to actually believe. Maybe only systems that have an explicit model? – Jack M Sep 14 '18 at 08:39
  • @JackM: You could use the diagonalization lemma to produce an arithmetic sentence saying "The theory that has me as its only axiom is consistent". Then take that sentence as the only axiom of a theory. Since the theory doesn't have any of the usual axioms that restrict how the $+$ and $\times$ symbols can behave, it will most likely be a simple task to produce a finite model for it -- but the details will depend on exactly how you've chosen to formalize statements about provability and consistency. – hmakholm left over Monica Sep 14 '18 at 08:49
  • @Peter Is Presburger arithmetic just the theory of $(\mathbb N,+)$? How would you even express consistency in that language? – bof Sep 14 '18 at 11:42
  • @bof That is what I wonder. But I would not have an idea how to express it in PA, for example, to be honest. – Peter Sep 14 '18 at 11:45
  • @Peter It cannot express its own consistency in any meaningful way. In fact, it can't even work with finite sequences! – Noah Schweber Sep 14 '18 at 17:02
  • @NoahSchweber I am curious how we then know that Presburg-arithmetic is consistent and complete. I guess a meta theory is necessary to prove this. But how can we proof that this meta theory is consistent etc. Why don't we run in an infinite loop like in the case of Goedels second incompleteness theorem ? – Peter Sep 14 '18 at 21:56
  • @Peter I mean, everything has to be proved in some background theory. Do you extend the same skepticism to, say, Godel's theorem itself? (What if the meta-theory we prove it in is inconsistent?) Or the infinitude of primes? Or ... Certainly the following is completely uncontroversial: the theory PA proves "Presburger arithmetic is complete and consistent." And PA can be replaced with vastly weaker theories here. We also have an explicit decision procedure for it. Etc. – Noah Schweber Sep 14 '18 at 23:07

0 Answers0