Comment by azan_
8 months ago
> As soon as Gödel published his first incompleteness theorem, I would have thought the entire field of mathematics would have gone full throttle on trying to find more axioms.
But why? Gödel's theorem does not depend on number of axioms but on them being recursively enumerable.
Right, Hilbert’s goal was (loosely speaking) to “find a finitely describable formal system” sufficient to “capture all truths”. When Gödel showed that can’t be done, that shouldn’t imply we just stop with the best theory we have so far and call it a day—it means there are an infinite number of more powerful theories (with necessarily longer minimal descriptions) waiting to be discovered.
In fact, both Gödel and Turing worked on this problem quite a bit. Gödel thought we might be able to find some sort of “meta-principle” that could guide us toward discovering an ever increasing hierarchy of more powerful axioms, and Turing’s work on ordinal progressions followed exactly this line of thinking as well. Feferman’s completeness theorem even showed that all arithmetical truths could be discovered via an infinite process. (Now of course this process is not finitely axiomatizable, but one can certainly extract some useful finite axioms out of it — the strength of PA after all is equivalent to the recursive iteration up to ε_0 of ‘Q_{n+1} = Q_n + Q_n is consistent’ where Q_0 is Robinson arithmetic).
Gödel's theorem shows that you need an infinite number of axioms to describe reality (given that available reality isn't finite), so any existing axiomatic system isn't enough.
Well, obviously we could simply take every true sentence of Peano arithmetic as an axiom to obtain a consistent and complete system, but if we think in that spirit, then almost every mathematician in the world is working on finding a better set of axioms (because every proof would either give us new axiom or show that something should not be included as axiom), right?
> obviously we could simply take every true sentence of Peano arithmetic as an axiom to obtain a consistent and complete system
If you’re talking about every true sentence in the language of PA, then not all such sentences are derivable via the theory of PA. If you are talking about the theorems of PA, then these are missing an infinite number of true statements in the language of PA.
Harvey Friedman’s “grand conjecture” is that virtually every theorem that working mathematicians actually publish can already be proved in Elementary Function Arithmetic (much weaker than PA in fact). So the majority of mathematicians are not pushing the boundaries of the existing foundational theories of mathematics, although there is certainly plenty of activity regardless.