What are some natural arithmetical statements independent of ZFC?

set-theory number-theory meta-math

Gödel's first incompleteness theorem produces a statement in the language of arithmetic that's independent of a given theory. The second theorem says that a consistient theory can not prove its own consistency, which is also a arithmetical statement (since you phrase it in terms of a Turing machine that looks for contradictions, for example).

Are there any "natural" statements in arithmetic that are independent of ZFC (besides consistency of ZFC, which is arguably pretty natural)? The wikipedia article only lists the consistency of ZFC.


Solution 1:

Well, "natural" is obviously very subjective, but Harvey Friedman has done a lot of work recently on innocuous-seeming combinatorial principles which imply the consistency of large cardinals.

He's organized these by templates which generate a number of such statements within a given topic. He has a number of manuscripts on these topics, including

  • Boolean relation theory

  • Strong Ramsey theorems for finite trees

  • Invariant maximality

You may also be interested in his numerous posts to the Foundations of Mathematics mailing list, the archives of which are searchable.

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