By Cohen's results, there exist models N and N' both satisfying the axioms of ZF without the Axiom of Choice such that the sentence θ expressing the Axiom of Choice for continuum-size families of subsets of the reals holds in N but not in N'.
Continuum-size(as a description of the scale being discussed)
Referring to infinity in the size of all real numbers (like all decimal points and fractions on a number line); it's the largest kind of infinity most people encounter in basic math.
Models (in mathematical logic)(as systems that satisfy axioms)
Different possible mathematical universes or systems that follow certain rules; think of them as alternate realities where different things might be true.
Sentence (in logic)(as a formal logical statement)
A mathematical statement that can be either true or false; in this case, θ (theta) is a sentence that represents the Axiom of Choice.
ZF (Zermelo-Fraenkel)(as a foundational axiom system)
A standard set of basic rules that mathematicians use to build all of mathematics from the ground up, without making any extra assumptions.
axioms(Stumpf, 1891)
Propositions that we assume to be true and necessary, originating in the content of judgments.
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)))