Worked solution: Gödel and Cohen: the continuum hypothesis is independent of ZFC (1963)

Step 9 of 9: Coda: after

\mathrm{CH}

— Easton's theorem and the search for new axioms

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In plain words

Once mathematicians knew the size of the continuum could be moved from $\aleph_1$ to $\aleph_2$ by forcing, a natural next question was: is there any limit at all to how large $2^{\aleph_0}$ (or $2^{\aleph_\alpha}$ for other cardinals $\alpha$ ) could consistently be? William Easton showed, only a few years after Cohen, that the freedom is enormous.

Yet decades later, Saharon Shelah discovered the story is not one of total freedom everywhere: at singular cardinals (limits of shorter sequences of smaller cardinals), hidden ZFC-provable constraints reappear, in a theory called PCF theory — a reminder that the independence phenomenon, however striking, does not mean set theory has nothing definite left to say about infinite cardinal arithmetic.

2^{\aleph_\alpha} \text{ can consistently be almost any regular cardinal} \ge \aleph_{\alpha+1} \text{ (Easton, 1970)}

Detailed analysis

Easton's theorem (1970) generalizes Cohen's method to force the continuum function $\alpha \mapsto 2^{\aleph_\alpha}$ (restricted to regular cardinals $\aleph_\alpha$ ) to take on almost any prescribed values consistent with the only ZFC-provable constraints known at regular cardinals: monotonicity and König's theorem ( $\mathrm{cf}(2^{\aleph_\alpha}) > \aleph_\alpha$ ). This shows the freedom Cohen discovered for $\aleph_0$ extends, in a precise sense, to every regular cardinal at once.

Saharon Shelah's PCF (possible cofinalities) theory, developed from the 1970s through the 1990s, showed that singular cardinals behave very differently: for example, ZFC itself proves that if $2^{\aleph_n} < \aleph_\omega$ for every finite $n$ , then $2^{\aleph_\omega} < \aleph_{\omega_4}$ — a genuine, non-forcing-avoidable upper bound with no counterpart at regular cardinals. This was a striking rebuttal to any hope that "everything about cardinal arithmetic is independent."

The search for definitive new axioms to settle $\mathrm{CH}$ itself continues today: forcing axioms like the proper forcing axiom favor $2^{\aleph_0} = \aleph_2$ , while Hugh Woodin's program (including his "Ultimate $L$ " project) has explored routes that would instead vindicate $\mathrm{CH}$ or a strong failure of it, depending on the decade of the program. No consensus exists, and Peter Koellner's Stanford Encyclopedia survey remains a good starting point for tracking where this open-ended story stands.

Terms in this step

Singular cardinal: An infinite cardinal that is the limit of a shorter sequence of smaller cardinals (such as $\aleph_\omega = \sup_n \aleph_n$ ), as opposed to a regular cardinal, which cannot be reached this way; singular cardinals obey extra ZFC-provable arithmetic constraints that regular cardinals do not.

Knowledge used in this step

The ZFC axioms