Scott's Theorem: Measurable Cardinals and $V \neq L$

Scott's theorem (1961) demonstrates that measurable cardinals are incompatible with Gödel's constructible universe, providing one of the first connections between large cardinals and the structure of the set-theoretic universe.

Statement

Theorem7.2Scott's Theorem

If there exists a measurable cardinal, then $V \neq L$ . Equivalently, the axiom of constructibility $V = L$ implies there are no measurable cardinals.

Proof

Suppose $\kappa$ is measurable and $V = L$ . Let $U$ be a $\kappa$ -complete non-principal ultrafilter on $\kappa$ , and form the ultrapower $j: V \to M = \mathrm{Ult}(V, U)$ .

Since $V = L$ , we have $M = L$ as well (the ultrapower of $L$ is $L$ ). The embedding $j: L \to L$ is a non-trivial elementary embedding with critical point $\kappa$ .

Now, in $L$ , the ordinal $\kappa$ is definable as "the least measurable cardinal" (if one exists). By elementarity: $j(\kappa)$ is the least measurable cardinal in $M = L$ , so $j(\kappa) = \kappa$ . But $j(\kappa) > \kappa$ since $\kappa$ is the critical point. Contradiction. $\square$

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Significance

RemarkLarge Cardinals Transcend $L$

Scott's theorem shows that the existence of a measurable cardinal is a genuinely "anti-constructibility" assumption. It implies that if large cardinals exist, then the set-theoretic universe must contain sets that are not constructible in Gödel's sense. This was one of the first indications that large cardinal axioms have profound structural consequences.

Example$0^\sharp$ and the Constructible Universe

A related result: $0^\sharp$ (zero sharp) exists if and only if there is a non-trivial elementary embedding $j: L \to L$ . The existence of $0^\sharp$ implies that $L$ is "thin" — every uncountable cardinal is inaccessible in $L$ — and provides an exact characterization of when the constructible universe is a proper subclass of $V$ .

RemarkInner Model Theory

Scott's theorem motivated the development of inner model theory, which constructs canonical models accommodating large cardinals. The models $L[U]$ (for a measurable cardinal), $K^{DJ}$ (the Dodd-Jensen core model), and Mitchell's models for sequences of measures generalize $L$ to accommodate progressively stronger large cardinals while retaining fine structural properties.

Scott's Theorem: Measurable Cardinals and V≠LV \neq LV=L

Statement

Significance

Scott's Theorem: Measurable Cardinals and $V \neq L$