User blog:Zetapology/alpha-type Variants

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In large cardinal axioms, elementarity is often used. For example, many weak large cardinal axioms like correct cardinals, reflecting cardinals, $0$-extendible cardinals, and uplifting cardinals can be defined using elementary substructures. However, these are easily strengthened by using infinitary logic in the elementarity. The result of this is the $\alpha$-type variant.

$R_\alpha$

Because the infinitary logic $\mathcal{L}_{\beta,\beta}$ only accepts regular $\beta$, the $\alpha$-th infinite regular ordinal $R_\alpha$ is used instead so the definition makes sense for all $\alpha$. An interesting result of doing this is that weakly inaccessible cardinals often appear because $R_\alpha=\alpha$ iff $\alpha$ is weakly inaccessible.

$$\omega_\alpha\;(\mathrm{if}\;\alpha=\kappa+n\;\mathrm{where}\;n<\omega\land\kappa\;\mathrm{is}\;\mathrm{weakly}\;\mathrm{inaccessible})$$ $$\omega_\alpha\;(\mathrm{if}\;\alpha<\omega)$$ $$\omega_{\alpha+1}\;(\mathrm{if}\;\mathrm{otherwise})$$

This definition is very important, and thus the consistency strength of $\alpha$-type variants relies on AC.

Definitions of Variants

The definitions of the variants are usually the original axioms except every instance of elementarity is replaced with $\mathcal{L}_{R_\alpha,R_\alpha}$ elementarity.

$\alpha$-type Correct cardinals

Let a cardinal $\kappa$ be $\alpha$-type correct iff $V_\kappa\prec_{\mathcal{L}_{R_\alpha,R_\alpha}}V$. Let $\kappa$ be supercorrect iff it is $\kappa$-type correct.

In terms of size, here is where these lay relative to eachother:

  • Every $\alpha$-type correct cardinal is $\beta$-type correct for every $\beta<\alpha$
  • If $\kappa$ is $\alpha$-type correct, then for every $\beta<\alpha$, $V_\kappa\models\exists\lambda(\lambda\;\mathrm{is}\;\beta\mathrm{-type}\;\mathrm{correct})$ (Therefore $\alpha$-type correctness is stronger than $\beta$-type correctness for any $\beta<\alpha$)
  • No cardinal $\kappa$ is $\beta$-type correct for any $\beta$ such that $R_\beta>\kappa$.
  • If $\kappa$ is supercorrect, then $V_\kappa\models\forall\beta\exists\lambda(\lambda\;\mathrm{is}\;\beta\mathrm{-type}\;\mathrm{correct})$
  • The smallest $\alpha$-type correct cardinal is less than the smallest $\alpha+1$-type correct cardinal (if both exist).

In terms of size, here is where these lay relative to other large cardinals:

  • The $0$-type correct cardinals are precisely the correct cardinals
  • The $1$-type correct cardinals are worldly. For any first-order theory $T$, $V\models T\Leftrightarrow V_\kappa\models T$.
  • If $\kappa$ is supercorrect, then it is reflecting and there is a reflecting cardinal in the rank of $\kappa$. It is not proven but conjectured that they are (strongly) Mahlo.
  • The least reflecting cardinal is not supercorrect.