The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture

Summary

§1. A tale of two problems. The formal independence of Cantor' Continuum Hypothesis from the axioms of Set Theory (ZFC) is an immediate corollary of the following two theorems where the statement of the Cohen's theorem is recast in the more modern formulation of the Boolean valued universe.

Theorem 1 (Gödel, [3]). Assume V = L. Then the Continuum Hypothesis holds.

Theorem 2 (Cohen, [1]). There exists a complete Boolean algebra, B, such that

V^B ⊨ “The Continuum Hypothesis is false”.

Is this really evidence (as is often cited) that the Continuum Hypothesis has no answer?

Another prominent problem from the early 20th century concerns the projective sets, [8]; these are the subsets of ℝⁿ which are generated from the closed sets in finitely many steps taking images by continuous functions, f : ℝⁿ → ℝⁿ, and complements. A function, f : ℝ → ℝ, is projective if the graph of f is a projective subset of ℝ × ℝ. Let Projective Uniformization be the assertion:

For each projective set A ⊂ ℝ × ℝ there exists a projective function, f : ℝ → ℝ, such that for all x ∈ ℝ if there exists y ∈ ℝ such that (x, y) ∈ A then (x, f(x)) ∈ A.

The two theorems above concerning the Continuum Hypothesis have versions for Projective Uniformization. Curiously the Boolean algebra for Cohen's theorem is the same in both cases, but in case of the problem of Projective Uniformization an additional hypothesis on V is necessary. While Cohen did not explicitly note the failure of Projective Uniformization, it is arguably implicit in his results.