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I was just idly thinking about things people have a hard time proving, like P=NP, etc, and wondering if instead it could be proved that it's provable or unprovable.

Is that a thing? Does that ever happen?

Antonio Vargas
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temporary_user_name
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    See https://en.wikipedia.org/wiki/Halting_problem – John Douma Sep 21 '15 at 20:22
  • If I understand correctly, you're providing this as an example of someone proving that something can't be proved, right? – temporary_user_name Sep 21 '15 at 20:24
  • Yes. This is a counter-example to the Entscheidungsproblem. – John Douma Sep 21 '15 at 20:29
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    Look up the continuum hypothesis. – Mirko Sep 21 '15 at 20:38
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    Gödels incompleteness theorem states that in fact the vast majority of true facts can't be proven! – Pax Sep 21 '15 at 21:08
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    @Chanler : $:$ What version of that says anything about the density of $\hspace{2.06 in}$ provable true facts vs. unprovable true facts? $;;;;$ –  Sep 24 '15 at 17:28
  • Axiom of choice or Zorn;s lemma. – Kushal Bhuyan Dec 10 '15 at 13:22
  • An example is Gauss, who proved that the Euclid's 5th axiom cannot be proved using the other 4 using hyperbolic geometry, a model for geometry where the first 4 axioms hol but not the 5th. This is an example where the constructs were thoroughly geometric, so not a logic/set theory example. – Balarka Sen Dec 10 '15 at 13:29
  • If you prove something to be provable, won't you just be proving it? – Aditya Agarwal Dec 10 '15 at 13:33
  • @AdityaAgarwal: My thought exactly! Constructivists might disagree (I am open to correction on this), but mainstream mathematics would accept such a proof. – TonyK Dec 10 '15 at 14:16
  • @AdityaAgarwal Intuitionists (e.g. Brouwer) don't agree that they have proved something just by showing it's provable. They don't believe in proofs without explicit construction (e.g., proof by contradiction). – Balarka Sen Dec 10 '15 at 21:15
  • @BalarkaSen An example would be appreciated. – Aditya Agarwal Dec 11 '15 at 03:19

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It is something mathematicians sometimes do, particularly set theorists and logicians. There is, of course, Goedel's result that no theory can prove its own consistency. There are also results that exploit this to make other provability statements. An example from my own field is that New Foundations set theory can't prove that something called the Axiom of Counting, because if it did it would prove the consistency of NF; we conclude that it is either inconsistent with NF, or a proper strengthening of the theory. Similarly, independence results are exactly theorems stating that neither a statement nor its negation can be proven from a set of axioms.

Strangely, I can't think of any examples of proving that something is provable that don't actually involve proving it, though in principle there's no reason that couldn't happen.

Malice Vidrine
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    About your last remark; I think that proving that something is provable requires one to show the existence of a proof. That in itself constitutes a proof of that something. – Servaes Dec 10 '15 at 13:29
  • @Servaes: I meant a proof in the object language. I may be able to show in our meta-theory that something holds of all groups (or whatever) and that this something is a first order property without ever writing a proof in the language $\mathcal{L}={e,-^{-1},\cdot}$. – Malice Vidrine Dec 10 '15 at 17:10
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A simple example. In the theory of groups, the proposition $$ \forall x\;\forall y\quad xy=yx $$ cannot be proved. The way to show this is to exhibit a nonabelian group.

In fact, when we say "can be proved" or "cannot be proved" we mean in a certain theory. And it makes no sense to say it without somehow having that theory in mind.

GEdgar
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