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I guess Goldbach's conjecture is a good example of a short open problem in number theory, and "Goodstein sequences reach 1" is a good example of a statement undecidable from first-order Peano arithmetic.

How about statements that are short in the language/alphabet of ZFC, though? Or in this alphabet extended by the standard abbreviations for null set, union, intersection, complement, and power set? My experience is that most statements that are short to write using arithmetic symbols +, $\cdot,$ ^ etc. become rather lengthy if written out without these abbreviations.

The axiom of choice is of course undecidable in ZF, and the continuum hypothesis/generalized continuum hypothesis is, in turn, undecidable in ZFC. When large cardinal axioms are written out, are they relatively short statements? And how about open problems? Do any of these refer (more or less) to finite sets?

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P=("Goldbach's conjecture is true" iff "Goodstein sequences reach 1") –  outsider Jul 20 '11 at 15:19

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Maybe the "union-closed sets conjecture" could be written as a short(ish) open problem in ZFC. Quoting Wikipedia,

A family of sets is said to be union-closed if the union of any two sets from the family remains in the family. The conjecture states that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family.

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Thank you! That's right along the lines of what I was thinking of existing. –  Daniel Briggs Jul 21 '11 at 4:19

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