Georg Cantor’s conjecture, the Continuum Hypothesis
Without equations, this states that for any set of real numbers, S, one of three things happen:
- S is finite.
- S has a 1-1 correspondence to the integers.
- S has a 1-1 correspondence to the reals.
There is nothing in between the integers and reals. Kurt Goedel showed that this was consistent with set theory, and Paul Cohen showed that its negation was consistent. In other words, CH is an undecidable proposition of Zermelo-Frankel set theory (and of ZFC, ZF with the Axiom Of Choice). Ditto for the Generalized Continuum Hypothesis. — Eric Jablow