The Continuum Hypothesis and relative consistency

In ZFC, CH is the equation 2^aleph-zero=aleph-one; GCH states 2^κ=κ-plus for every infinite cardinal κ. Gödel's constructible interpretation shows that consistency of ZF implies consistency of ZFC plus GCH, and hence of ZFC plus CH. Cohen's forcing theorem supplies the other direction needed for independence: consistency of ZFC implies consistency of ZFC plus not-CH. Neither theorem proves that its antecedent is true, and an inconsistent starting theory would make the apparent alternatives meaningless.

Assuming ZFC is consistent, the two relative consistency results imply that ZFC proves neither CH nor its negation. If it proved one, the consistent extension by the opposite sentence would inherit a contradiction. A familiar exposition starts with an externally countable transitive M satisfying ZFC plus GCH, but such a model is a stronger hypothesis than mere consistency. The general metatheorem is justified through formal forcing methods or finite-fragment arguments. It is not obtained by claiming that every countable model can be collapsed into an externally well-founded one.

For the concrete conditional construction, take such an M and let κ be its aleph-two. Inside M use finite partial functions from κ×ω to two, ordered by reverse inclusion. A generic union defines a real cα at each coordinate α<κ. For any two different coordinates, the conditions making their bits differ somewhere are dense: choose a fresh natural position and assign opposite bits. Thus the extension contains at least κ pairwise distinct coordinate reals, with the quantifiers and cardinal labels interpreted in the indicated models.

The forcing has the countable chain condition. An uncountable family of finite domains has an uncountable delta-system subfamily with one finite root; thinning again makes the bit assignments on the root agree. Two remaining conditions then have a common union, contradicting an antichain. The standard ccc preservation theorem, cited here, preserves cardinals and cofinalities. In particular the ground-model κ remains aleph-two in the extension. Without this preservation step, a claimed supply of κ different reals could coexist with κ having become a smaller cardinal.

To obtain the exact continuum, use nice names for subsets of ω. Such a name is determined by one antichain of conditions for each natural number, indicating which conditions force that number into the subset. Every real has a nice name, and ccc makes these antichains countable in M. Since P has cardinality κ in M, the number of nice names is at most κ^aleph-zero as computed there. Counting arbitrary names instead would fail: a single set can have a proper class of redundant names internally.

Ground-model GCH gives κ^aleph-zero=κ for κ=aleph-two. The name bound therefore gives continuum at most κ in the extension, and the distinct coordinate reals give the reverse inequality. All these cardinal calculations are internal to the relevant model; external countability of M does not turn its internal aleph-two into its internal aleph-zero. This construction establishes one permitted continuum value. It does not imply that every larger cardinal is allowed, because the cofinality restrictions proved earlier continue to constrain cardinal exponentiation.