Critical point

• Critical Point
• Formal Definition
• Use in Large Cardinal Axioms

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 Reinhardt cardinal, which is inconsistent with the Axiom of Choice. Thus, the following axioms mention critical points:

• Rank into rank cardinals (axioms I0-I3)
• The wholeness axiom
• Huge cardinals:
  • almost n-huge
  • almost super-n-huge
  • n-huge
  • super-n-huge
  • $\omega$-huge
• High jump cardinals:
  • almost high-jump
  • high-jump
  • super high-jump
  • high-jump with unbounded excess closure
• $\alpha$-extendible
• Compact cardinals:
  • $\lambda$-supercompact
  • strongly compact
  • nearly supercompact
• Strong cardinals:
  • superstrong
  • $\theta$-strong
• Tall cardinals