## Critical Point Given two structures $\mathcal{M}$ and $\mathcal{N}$, loosely speaking, the *critical point* is the smallest element in $\mathcal{M}$ which is similar to a larger element in $\mathcal{N}$. The actual definition can only be told if both structures' universes are transitive classes containing ordinals. ## Formal Definition Given two transitive classes $\mathcal{M}$ and $\mathcal{N}$, and an elementary embedding $j:\mathcal{M}\rightarrow\mathcal{N}$, the critical point of $j$ (often denoted $\mathrm{cp}(j)$) is the smallest ordinal $\alpha$ in $\mathrm{M}$ such that $\alpha\neq j(\alpha)$. ## Use in Large Cardinal Axioms Critical points are used in Large Cardinal Axioms to make very large $\mathrm{M}$ such that $j:V\rightarrow\mathcal{M}$ is an elementary embedding. The closer $\mathcal{M}$ gets to $V$, the closer one is to proving the Wholeness Axiom. Assuming this embedding has a critical point, one gets closer to a <a href="Reinhardt" class="mw-redirect" title="Reinhardt">Reinhardt</a> cardinal, which is inconsistent with the Axiom of Choice. Thus, the following axioms mention critical points: - [](Rank_into_rank.md) cardinals (axioms I0-I3) - The [wholeness axiom](Wholeness_axiom "Wholeness axiom") - Huge cardinals: - [](Huge.md) - [](Huge.md) - [n-huge](Huge.md "Huge") - [super-n-huge](Huge.md "Huge") - [$\omega$-huge](Huge.md "Huge") - High jump cardinals: - [](High-jump.md) - [high-jump](High-jump.md "High-jump") - [](High-jump.md) - [](High-jump.md) - [$\alpha$-extendible](Extendible.md "Extendible") - Compact cardinals: - [$\lambda$-supercompact](Supercompact.md "Supercompact") - [](Strongly%20compact.md) - [](Nearly%20supercompact.md) - Strong cardinals: - [superstrong](Superstrong.md "Superstrong") - [$\theta$-strong](Strong.md "Strong") - [Tall](Tall.md "Tall") cardinals