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Recall that Gödel's First Incompleteness Theorem (denote GIT1) states, roughly, any axiomatic system $S$ stronger than Peano Arithmetic (PA) cannot be "covered" by finitely many axioms. GIT2 states, roughly, that furthermore such $S$ cannot prove, within itself, its own consistency (denote $\mathrm{cons}\; S$). And it is often said that GIT1, together with GIT2, invalidates Hilbert's program (HP).

It's rather obvious to me that GIT1 declares HP to be problematic, if not impossible. But what GIT2 says of HP, is less obvious. Indeed, even if $\mathrm{cons}\; S$ were a theorem within S, that is

$$\models_S \mathrm{cons}\; S$$

and suppose we do not know GIT2. Then, it is possible that $S$ is inconsistent, and a contradiction formula is a theorem in $S$, which in turn entails $\mathrm{cons}\; S$. Thus, it seems, whether $\mathrm{cons}\; S$ is a theorem or not, is not very important. If so, GIT2 does not really tell us much by saying that $\mathrm{cons}\; S$ is not a theorem in $S$.

So how was GIT2 a bad news to Hilbert's view?

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First, as others have pointed out, your characterization of GIT1 is not right. What it actually roughly says is that any complete and consistent theory stronger than PA cannot be generated by a decidable set of axioms. This does kill the prospects of a single formal system that decides “all of mathematics” which was one of Hilbert’s loftier goals.$^*$

But another big aspect was to have a finitary proof on the consistency of such a system that underlies mathematics, which still makes sense as a goal even if that system is necessarily incomplete. For example. we could say the system is ZFC, which we know allows us to formalize most of known mathematics, even if we also know there are statements it can’t decide (provided it is consistent).

I have a few qualms with the argument that a proof of Con(S) in S would not tell us much, but for the most part I agree, so I’ll grant it for the purposes of this answer. However, the goal is much more ambitious than this: we want to prove the consistency of our system of mathematics (say ZFC) in a very weak finitary theory of arithmetic (say PRA), not merely in ZFC. Surely it would be desirable and meaningful if that could be carried out. However, if GIT2 tells us ZFC can’t prove it, then certainly a weaker theory like PRA can’t either. So GIT2 is indeed bad news for this aspect of the program.

$^*$ I’m a lousy mathematician but an even worse historian. What I say is based on my understanding from the folklore of what Hilbert’s Program was and may or may not correlate with anything Hilbert actually said or believed.

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Hilbert's Program was a search for a solid foundation for mathematics in which two obvious facts about mathematics hold true. Every statement can be proven true or false and every result proved is non-trivially true. If $S$ is inconsistent one can prove any statement is true using proof by contradiction, meaning proving a statement is true in an inconsistent axiom system is meaningless.

What Gödels second incompleteness theorem shows is that we can never show that any "interesting" axiom system is consistent, meaning that any day a mathematician can show a result which contradicts previous results and render years of mathematical progress worthless. (For mathematicians that is, the rest of society can still happily use the mathematics on which it is build.)

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  • $\begingroup$ But if we question S's consistency, why did Hilbert aim to show cons S within S? I mean, even if we show cons S, we are still not sure of S's consistency. $\endgroup$ Feb 19 '19 at 11:48
  • $\begingroup$ Welcome to Math.SE! There are many things written about the 2nd incompleteness theorem, but it does not prevent us from proving consistency. For example both Gentzen and Goedel proves the consistency of Peano Arithmetic. The second incompleteness theorem does prevent us from proving the consistency of strong systems using only finitistic means, of course, which does impact Hilbert's program. But that does not stop us proving consistency at all. $\endgroup$ Feb 19 '19 at 11:57
  • $\begingroup$ You're probably aware of that, of course, but on a site like this the readers often take statements in answers too literally. $\endgroup$ Feb 19 '19 at 12:06
  • $\begingroup$ "Can show a result which contradicts previous results and render years of mathematical progress worthless." It probably would not be that bad. We could probably still use the ideas from most of the wrong proofs. $\endgroup$
    – PyRulez
    Feb 19 '19 at 13:10
  • $\begingroup$ True, I exaggerated a little bit. Though I would argue if one of the well-established axiom systems is found to be inconsistent it would cause a lot of problems to fix it, if only because all mathematicians need to develop a completely new intuition. $\endgroup$ Feb 19 '19 at 16:44
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I think the question "Usefulness of Gödel's incompleteness theorem" actually is a duplicate of my question, and the answer there is satisfying.

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  • $\begingroup$ Well, Mauro ALLEGRANZA has good intention, but what he/she cites seems to me somewhat beside the point I made. $\endgroup$ Feb 19 '19 at 17:46

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