The truth of the second-order sentence θ_CH in a sufficiently large model depends on whether the Continuum Hypothesis holds, and the Continuum Hypothesis is not settled by ZFC alone.
(Classical set theory as a foundation for mathematics)
The axiom system ZF plus the axiom of choice (AC).
model(Possible worlds interpretation of S5 adapted for modal nonmonotonic logic)
A pair <I, S> where I is a set of literals (a state description / possible world) and S is a set of complete, consistent sets of literals (interpretations) with I ∈ S
second-order sentence(in logic and mathematics)
A logical statement that doesn't just talk about objects (like 'cats are animals'), but talks about properties or sets of objects themselves (like 'some properties are shared by all mammals'). It's a step up in complexity from regular statements.
θ_CH(Second-order logic semantics)
A sentence of the empty vocabulary in second-order logic that has a model if and only if the Continuum Hypothesis holds.
Let \(\theta_{\le}(P,R)\) be the formula \[ \exists F\left(\forall x\,\forall y\left( (F(x)=F(y)\to x=y) \land(P(x)\to R(F(x)) \right)\right). \] Now \(\mm\models_s\theta_\le(P,R)\) if and only if \(|s(P)|\le |s(R)|\). Let \(\theta_{\textrm{EQ}}(P,R)\) be the formula \(\theta_{{\le}}(P,R)\land \theta_{{\le}}(R,P)\). Now \(\mm\models_s\phi(P,R)\) if and only if \(|s(P)|=|s(R)|\). Let \(\theta'_{\textrm{EC}}(Y)\) be \[ \exists F\left( \forall x\,\forall y((F(x)=F(y)\to x=y)\land R(F(x)))