\(A_6^*\) is equivalent to \(A_6\) [WIP]

In there paper Strong Axioms of Infinity and Elementary Embedding, Robert Solovay, William Reinhardt, and Akihiro Kanamori introduced the axioms \(A_1\)-\(A_7\), measuring the gap between hugeness and extendibility.

\(A_1(\kappa)\): \(\kappa\) is huge.

\(A_2(\kappa)\): There is some \(j: V_\alpha\prec V_\beta\) with critical point \(\kappa\) such that \(j(\kappa)\le\alpha\).

\(A_3(\kappa)\): \(\kappa\) is almost huge; i.e. there is some \(j: V\prec M\) with critical point \(\kappa\) such that \(M^\lambda\subseteq M\) for every \(\lambda\lt j(\kappa)\); we call this \(M^{\lt j(\kappa)}\subseteq M\).

\(A_4(\kappa)\): There is some \(\lambda\gt\kappa\) and a normal ultrafilter \(U\) over \(P_\kappa(\lambda)\) such that if \(f: \kappa\rightarrow\kappa\), then \(j_U(f)(\kappa)\lt\lambda\).

\(A_5(\kappa)\): There is a normal ultrafilter \(U\) over \(\kappa\) such that for any natural sequence of structure \(\langle\mathfrak M_\zeta|\zeta\lt\kappa\rangle\) each in \(V_\kappa\), there is some \(X\in U\) such that if \(\alpha\lt\beta\in X\) there is some \(j: \mathfrak M_\alpha\prec\mathfrak M_\beta\) such that \(\text{crit}j=\alpha\).

\(A_6(\kappa)\): \(V_{\kappa+1}\) is a natural model of \(KM\) set theory \(+\)Vopěnka’s principle.

\(A_6^*(\kappa)\): \(V_{\kappa+1}\) is a natural model of \(KM\) set theory and there is a class \(S\), stationary in \(\kappa\), such that if \(\alpha_0\lt…\lt\alpha_n\lt\beta_0\lt…\lt\beta_n\in S\), then there is some \(j: V_{\alpha_n}\prec V_{\beta_n}\) such that \(j(\alpha_i)=\beta_i\).

\(A_7(\kappa)\): \(\kappa\) is extendible and there is a normal ultrafilter over \(\kappa\) such that \(\{\alpha\lt\kappa|\alpha\text{ is extendible}\}\in U\).

If \(A_1(\kappa)\), there is a normal ultrafilter \(U\) over \(\kappa\) such that \(\{\alpha\lt\kappa|A_2(\alpha)\}\in U\); if \(A_2(\kappa)\), then \(A_3(\kappa)\) and there is a normal ultrafilter \(U\) over \(\kappa\) such that \(\{\alpha\lt\kappa|A_3(\alpha)\}\in U\); and if \(A_3(\kappa)\), then \(A_4(\kappa)\) and there is a normal ultrafilter \(U\) over \(\kappa\) such that \(\{\alpha\lt\kappa|A_4(\alpha)\}\in U\).

The first part is simple; take some \(j: V\prec M\) witnessing the hugeness of \(\kappa\); then \(|j\restriction V_{j(\kappa)}|=|V_{j(\kappa)}|=j(\kappa)\) and so \(j\restriction V_{j(\kappa)}: V_{j(\kappa)}\prec V_{j^2(\kappa)}^M\) witnesses \(A_2(\kappa)\) in \(M\), and so \(\{\alpha\lt\kappa|A_2(\alpha)\}\in\{X\subseteq\kappa|\kappa\in j(X)\}\).

The second and third parts are trickier, but with manipulation of sequences of ultrafilters and ultrafilters respectively, we can get the result we want.

We call a cardinal \(\kappa\) Vopěnka if \(A_6(\kappa)\). If \(\kappa\) is Vopěnka and \(U\) is a normal ultrafilter over \(\kappa\), then \(\{\alpha\lt\kappa|\alpha\text{ is Vopěnka}\}\in U\), because \(V_{\kappa+1}\subseteq \text{Ult}_U(V)\) and the Vopěnkaness of \(\kappa\) is a property of \(V_{\kappa+1}\). In particular, if \(U\) witnesses \(A_5(\kappa)\), then \(\{\alpha\lt\kappa|\alpha\text{ is Vopěnka}\}\in U\).

Furthermore, if \(\alpha\in S\) and \(S\) witnesses \(A_6^*(\kappa)\), then \(V_\alpha\prec V_\kappa\) and \(A_7(\alpha)\); i.e. \(\alpha\) is extendible and there is a normal ultrafilter over \(\kappa\) such that \(\{\beta\lt\alpha|\beta\text{ is extendible}\}\in U\).

Proposition: \(A_6^*(\kappa)\) if and only if \(\kappa\) is Vopěnka.

\(Proof.\) For the forward direction, we use the equivalent definition of Vopěnkaness; \(\kappa\) is Vopěnka if and only if for every \(A\subseteq V_\kappa\), there is some \(\alpha\) such that for every \(\eta\lt\kappa\), there is some \(j: (V_{\kappa+\eta},\in,A\cap V_{\kappa+\eta})\prec (V_\zeta,\in,A\cap V_\zeta)\). Fix some \(A\subseteq V_\kappa\). Then \(C=\{\alpha\lt\kappa|(V_\alpha,\in,A\cap V_\alpha)\prec(V_\kappa,\in,A\cap V_\kappa)\}\) is club in \(\kappa\), because \(\kappa\) is inaccessible, and so fix any \(\alpha_0\in C\cap S\) and a sequence \(\alpha_0\lt\alpha_0+\eta\lt\alpha_1…\lt\alpha_n\lt\beta_0\lt…\lt\beta_n\in C\cap S\).

Then \((V_\kappa,\in,A)\vDash\phi(x)\) if and only if \((V_{\alpha_n},\in,V_{\alpha_n}\cap A)\vDash\phi(x)\) if and only if \((V_{\beta_n},\in,A_0)\vDash\phi(j(x))\) if and only if \((V_{\beta_n},\in,A\cap V_{\beta_n})\vDash\phi(j(x))\). Then \(j\restriction V_{\alpha+\eta}: (V_{\kappa+\eta},\in,A\cap V_{\kappa+\eta})\prec (V_\zeta,\in,A\cap V_\zeta)\).

For the converse direction, let \(F_{\text{Vop},\kappa}\) be the Vopěnka filter, and \(S_\sigma\) be defined by induction along the length of \(\sigma\), for \(\sigma\in\kappa^{\lt\omega}\). If \(|\sigma|=n+1\), \(n\) is even and \(\sigma(n)\in S_{\sigma\restriction n}\), then let:

\(S_\sigma=\{\alpha\lt\kappa|\exists j: V_{\sigma(n/2)}\prec V_{\sigma(n/2)}(\text{crit}j=\alpha_0\land\forall i\lt n(j(\sigma(i))=\sigma(i+n/2))\}\)

If \(n\) is odd:

\(S_\sigma=\{\alpha\lt\kappa|\exists j: V_{\sigma((n+1)/2)}\prec V_{\sigma((n+1)/2)}(\text{crit}j=\alpha_0\land\forall i\lt n(j(\sigma(i))=\sigma(i+(n+1)/2))\}\)

Else \(S_\sigma=\kappa\). We prove \(S_\sigma\in F_{\text{Vop},\kappa}\).

Norman Perlmutter introduced a series of large cardinals designed to measure the gap between almost hugeness and supercompactness. If \(j: V\prec M\) is an elementary embedding, then let the clear of \(j\) \(\theta=\text{sup}\{j(f)(\text{crit}j)|f: \text{crit}j\rightarrow \text{crit}j\}\). \(\kappa\) is high-jump if and only if there is some \(j: V\prec M\) such that \(M^\theta\subseteq M\) for \(\theta\) the clearance of \(j\) and \(\kappa\) the critical point \(j\). Respectively, \(\kappa\) is Shelah for supercompact if and only if for every function \(f: \kappa\rightarrow\kappa\), there is some \(j: V\prec M\) such that \(M^{j(f)(\kappa)}\subseteq M\) and \(\text{crit}j=\kappa\).

Proposition: If \(\kappa\) is Shelah for supercompactnes, then \(A_6(\kappa)\), and there is a normal ultrafilter \(U\) such that \(\{\alpha\lt\kappa|A_6(\kappa)\}\in U\).

\(Proof.\) By a result of Permultter, \(A_6(\kappa)\) if and only if \(\kappa\) is Woodin for supercompactness, and because \(\kappa\) is Shelah for supercompactness, \(\kappa\) is Woodin for supercompactness. Also, given any \(U\) a normal ultrafilter over \(\kappa\), \(V_{\kappa+1}\subseteq\text{Ult}_U(V)\) and so \(\text{Ult}_U(V)\vDash A_6(\kappa)\), and therefore \(\{\alpha\lt\kappa|A_6(\kappa)\}\in U\).■

Proposition: \(\kappa\) is \(2-\)fold Shelah if and only if \(\kappa\) is Shelah for supercompactness.

Proposition: If \(A_5(\kappa)\), as witnessed by \(U\), then \(\{\alpha\lt\kappa|\alpha\text{ is Shelah for supercompactness}\}\in U\).

Proposition: \(A_4(\kappa)\) if and only if \(\kappa\) is high-jump.

What about the lower reaches of Permultter’s hierarchy?

Proposition: \(\kappa\) is enhanced supercompact if and only if it is extendible and there is a strong cardinal above it.

Corollary: If \(A_7(\kappa)\), as witnessed by \(U\), then \(\{\alpha\lt\kappa|\alpha\text{ is enhanced supercompact}\}\in U\).

Proposition: \(\kappa\) is \((2^\kappa)^+-\)hypercompact if and only if it is hypercompact.

Proposition: If \(\kappa\) is extendible, it is hypercompact.

Corollary: If \(\kappa\) is enhanced supercompact, it is hypercompact.

Proposition: If \(\kappa\) is \(\beta+1-\)hypercompact and \(\beta\lt (2^\kappa)^+\), then \(\kappa\) is excessively \(\beta-\)hypercompact and there is a normal ultrafilter \(U\) such that \(\{\beta\lt\kappa|\beta\text{ is excessively }\beta-\text{hypercompact}\}\in U\).

Proposition: If \(\kappa\) is excessively \(\beta-\)hypercompact and \(\kappa^+\lt\beta\lt (2^\kappa)^+\), then there is a normal ultrafilter \(U\) such that \(\{\beta\lt\kappa|\beta\text{ is }\beta-\text{hypercompact}\}\in U\).