ⓘ Critical point, set theory. In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ord ..

                                     

ⓘ Critical point (set theory)

In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself.

Suppose that j: N → M {\displaystyle j:N\to M} is an elementary embedding where N {\displaystyle N} and M {\displaystyle M} are transitive classes and j {\displaystyle j} is definable in N {\displaystyle N} by a formula of set theory with parameters from N {\displaystyle N}. Then j {\displaystyle j} must take ordinals to ordinals and j {\displaystyle j} must be strictly increasing. Also j ω = ω {\displaystyle j\omega=\omega }. If j α = α {\displaystyle j\alpha=\alpha } for all α < κ {\displaystyle \alpha j κ > κ {\displaystyle j\kappa> \kappa }, then κ {\displaystyle \kappa } is said to be the critical point of j {\displaystyle j}.

If N {\displaystyle N} is V, then κ {\displaystyle \kappa } the critical point of j {\displaystyle j} is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a κ {\displaystyle \kappa } -complete, non-principal ultrafilter over κ {\displaystyle \kappa }. Specifically, one may take the filter to be { A ∣ A ⊆ κ ∧ κ ∈ j A } {\displaystyle \{A\mid A\subseteq \kappa \land \kappa \in jA\}}. Generally, there will be many other

                                     
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