Strengthened Formulae

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Cardinals Defined via Critical Point

Many large cardinal axioms are defined via critical point of an elementary embedding. For example, strong cardinals, tall cardinals, huge cardinals, measurable cardinals, etc. Most of those large cardinal axioms are defined specifically in this way:

$\kappa$ is $\theta$-Axiom when it is the critical point of some elementary embedding $j:V\rightarrow M$ such that $\phi(M,\kappa,j,\theta)$.

Given a formula $\phi$, a cardinal $\kappa$, and an ordinal $\theta$, we could let the following be defined as $F_\phi(\kappa,\theta)$:$$\exists M\exists j:V\rightarrow M (M\;\mathrm{is}\;\mathrm{a}\;\mathrm{transitive}\;\mathrm{inner}\;\mathrm{model}\;\mathrm{of}\;\mathrm{ZFC}\land j\;\mathrm{is}\;\mathrm{an}\;\mathrm{elementary}\;\mathrm{embedding}\land\mathrm{cp}(j)=\kappa\land\phi(M,\kappa,j,\theta))$$

Here is a list of $F_\phi(\kappa,\theta)$ equivalences for different $\phi$:

  • $\phi(M,\kappa,j,\theta)$ is $M=M$ : $F_\phi(\kappa,\theta)$ is "$\kappa$ is measurable".
  • $\phi(M,\kappa,j,\theta)$ is $V_\theta\subset M$ : $F_\phi(\kappa,\theta)$ is "$\kappa$ is $\theta$-strong".
  • $\phi(M,\kappa,j,\theta)$ is $j(\kappa)>\theta\land M^\kappa\subset M$ : $F_\phi(\kappa,\theta)$ is "$\kappa$ is $\theta$-tall"
  • $\phi(M,\kappa,j,\theta)$ is $V_{j^{\theta}(\kappa)}\subset M$ : $F_\phi(\kappa,\theta)$ is "$\kappa$ is $\theta$-superstrong"
  • $\phi(M,\kappa,j,\theta)$ is ${}^\theta M\subset M$ : $F_\phi(\kappa,\theta)$ is "$\kappa$ is $\theta$-supercompact"
  • $\phi(M,\kappa,j,\theta)$ is ${}^{j^{\theta-1}(\kappa)} M\subset M$ : $F_\phi(\kappa,\theta)$ is "$\kappa$ is $\theta$-huge"

Strengthening $F_\phi(\kappa,\theta)$ to $G_\phi(\kappa,\theta)$

Although these cardinals we have so far defined are somewhat strong, they are not as strong as they could be. We will construct a method of making these large cardinals much larger.

Given a formula $\phi$, a cardinal $\kappa$, and an ordinal $\theta$, we could let the following be defined as $G_\phi(\kappa,\theta)$:$$\exists M\exists j:M\rightarrow M (M\;\mathrm{is}\;\mathrm{a}\;\mathrm{transitive}\;\mathrm{inner}\;\mathrm{model}\;\mathrm{of}\;\mathrm{ZFC}\land j\;\mathrm{is}\;\mathrm{a}\;\mathrm{nontrivial}\;\mathrm{elementary}\;\mathrm{embedding}\land\mathrm{cp}(j)=\kappa\land\phi(M,\kappa,j,\theta))$$

With that, we strengthen the previous large cardinal axioms:

  • $\phi(M,\kappa,j,\theta)$ is $\exists\lambda(V_\lambda = M)$ : $G_\phi(\kappa,\theta)$ is "$\kappa$ is I3".
  • $\phi(M,\kappa,j,\theta)$ is $\exists\lambda(V_{\lambda+1} = M)$ : $G_\phi(\kappa,\theta)$ is "$\kappa$ is I1".
  • $\phi(M,\kappa,j,\theta)$ is $\exists\lambda(L(V_{\lambda+1}) = M)$ : $G_\phi(\kappa,\theta)$ is "$\kappa$ is I0".
  • $\phi(M,\kappa,j,\theta)$ is $V_\theta\subset M$ : $G_\phi(\kappa,\theta)$ implies that $\kappa\geq\lambda$ for some I3 cardinal $\lambda$ (strengthening of strong cardinal)
  • $\phi(M,\kappa,j,\theta)$ is $V_{j^{\theta}(\kappa)}\subset M$ : $G_\phi(\kappa,\theta)$ implies that $\kappa\geq\lambda$ for some cardinal $\lambda$ the critical point of an elementary embedding $j:V_\mu\rightarrow V_\mu$ such that $j^{\theta+1}(\lambda)=j^{\theta}(\lambda)$ (strengthening of $\theta$-superstrong cardinal)