[FOM] Refuting CH?

[Harvey Friedman]

GENERAL PRINCIPLE. Any simple existential property that holds of all
simple Borel data, in fact holds of all such simple data, regardless
of whether it is Borel or not.

"THEOREM". The General Principle is consistent and refutes CH.

So here is an example. We use n below for integers.

THEOREM 1. Let f:R into R be Borel. There exists x,y such that x is
not any f(y+n) and y is not any f(x+n).

PROPOSITION 2. Let f_1,f_2,... be functions from R into R. There
exists x,y such that x is not any f_i(y) and y is not any f_j(x).

THEOREM 3. Proposition 2 is equivalent to not CH.

Harvey Friedman