TITLE: Mathematical statement with simple independence proof from $\mathsf{ZF}$
QUESTION [19 upvotes]: Is it possible for someone with little set-theoretic knowledge (e.g., me) to understand the proofs that either $\mathsf{CH}$ or $\mathsf{AC}$ is independent of $\mathsf{ZF}$?
I am looking for any kind of mathematical-sounding statement (not "This sentence is unprovable") for which the proof of independence is somewhat accessible.

REPLY [3 votes]: If you're willing to drop from ZF(C) down to PA - that is, settle for arithmetic rather than sets - then I believe Goodstein's Theorem (or slightly more complicatedly, the Paris-Harrington theorem) might be what you're after; both of these are relatively straightforward statements of arithmetic.  The proofs of independence are unfortunately still fairly complicated and full of technical details, but the core concept ('Peano Arithmetic isn't strong enough to prove induction principles past a certain point') may be more straightforward to grok than the mechanics of forcing and building models of set theory.
As far as forcing itself goes, I think the most amateur-friendly approach may be through modal logic; I'm a great fan of Smullyan and Fitting's Set Theory and the Continuum Problem.  It has its share of (mostly printing) flaws, but it's really quite accessible and covers the material well.  You might want Smullyan's more casual Forever Undecided (if you can find it!) as a warm-up; it goes over some of the basics of modal logic in a puzzle-oriented setting.