
Hilbert's 24th problem - infinity0
https://en.wikipedia.org/wiki/Hilbert%27s_24th_problem
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simias
I don't understand why there's a "See also" link to the "23 enigma" which as
far as I can tell is not related at all to this 24th problem.

I'm also not sure it's really worthwhile to post a stub of a wikipedia article
on HN without any commentary or context... What is the relevance of this 24th
problem? Why is it interesting?

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heydenberk
Seems like this presages this idea of Kolmogorov[0] Complexity by half a
century.

[0]
[http://en.wikipedia.org/wiki/Kolmogorov_complexity](http://en.wikipedia.org/wiki/Kolmogorov_complexity)

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JadeNB
I don't know—there's a big gap between asking "how do you measure this
philosophical-sounding quantity?" and actually coming up with a concrete means
of doing so. I could ask here "how do you define consciousness for machines?",
but (even if I were the first to do so) it wouldn't be fair to say that I'd
anticipated Sharmach's landmark 2098 paper solving the problem.

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hiker
I think Kolmogorov complexity is spot on comparison, considering programs as
proofs, theorems as types in the Curry–Howard isomorphism.

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Grue3
Suppose it is possible to test whether a statement is an axiom in our theory
(this is true for finite sets of axioms, or computable infinite sets, i.e.
pretty much all sets of axioms used in reality). Suppose there is a finite
number of derivation rules (probably will work for computable set of
derivation rules). Then it is possible to check whether a statement is derived
from axioms in one step, i.e. the set of such statements is computable. By the
same reasoning, the set of statements that are derivable in two steps is also
computable. And so on. Thus the problem of whether the statement is derivable
in less than M steps (if M steps is the shortest known proof) is decidable.

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blablabla123
While doing my first Analysis for Mathematics course, I had big problems
coming up with proofs when I started. So I looked for Books on the topic,
didn't really find much (=nothing useful). Seems to be a quite untouched
topic...

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jackmaney
Most universities have a course specifically geared towards proofs, including
first order propositional and predicate logic, set theory, basic proof
techniques, etc. I've taught this particular course a few times when I was in
academia, and one of the better books that was around at the time (2004--8)
was Krantz's _Elements of Advanced Mathematics_.

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jackmaney
Also, the initial dialog in _Proofs and Refutations_ by Imre Lakatos is a nice
discussion not only of proofs, but on thought processes that one often goes
through during mathematical research.

