According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $\lambda$ and nontrivial elementary embedding $j:V_{\lambda +2}\to V_{\lambda +2}$. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered.

$I_{0,\lambda }$ is the assertion, introduced by Hugh Woodin, that $\lambda$ is an ordinal and there is an elementary embedding $j:L(V_{\lambda +1})\to L(V_{\lambda +1})$ with critical point $<\lambda$. And $I_0$ asserts that $I_{0,\lambda }$ holds for some $\lambda$. The axiom $I_0$ is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe $V$ (in which case $\lambda$ must be a limit ordinal), but we assume only ZF.

We prove, assuming ZF $+$$I_{0,\lambda }$$+$ “$\lambda$ is an even ordinal”, that there is a proper class transitive inner model $M$ containing $V_{\lambda +1}$ and satisfying ZF $+$$I_{0,\lambda }$$+$ “there is an elementary embedding $k:V_{\lambda +2}\to V_{\lambda +2}$”; in fact we will have $k$ ⊆$j$, where $j$ witnesses $I_{0,\lambda }$ in $M$. This result was first proved by the author under the added assumption that $V_{\lambda +1}^{\#}$ exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also $\lambda$ is a limit ordinal and $\lambda$-DC holds in $V$, then the model $M$ will also satisfy $\lambda$-DC.

We show that ZFC $+$ “$\lambda$ is even” $+$$I_{0,\lambda }$ implies $A^{\#}$ exists for every $A\in V_{\lambda +1}$, but if consistent, this theory does not imply $V_{\lambda +1}^{\#}$ exists.

根据 Kenneth Kunen 的定理，在 ZFC 下，不存在序数$\lambda$和非平凡的基本嵌入$j：V_{\lambda +2}\to V_{\lambda +2}$。他的证明依赖于选择公理（AC），目前尚未发现仅来自 ZF 的证明。

${我}_{0,\lambda }$休·伍丁 (Hugh Woodin) 提出这样的断言：$\lambda$是一个序数并且有一个基本嵌入$j：L（V_{\lambda +1}）\to L（V_{\lambda +1}）$有临界点$<\lambda$。和${我}_0$断言${我}_{0,\lambda }$对某些人来说成立$\lambda$。公理${我}_0$是已知与AC不符的最强大红衣主教之一。通常在假设 ZFC 在整个宇宙中的情况下进行研究$V$（在这种情况下$\lambda$必须是极限序数），但我们假设只有 ZF。

我们证明，假设 ZF$+$${我}_{0,\lambda }$$+$“$\lambda$是一个偶序数”，存在一个真类传递内部模型$\mathrm{中号}$含有$V_{\lambda +1}$并满足ZF$+$${我}_{0,\lambda }$$+$“有一个基本的嵌入$k：V_{\lambda +2}\to V_{\lambda +2}$”；事实上我们会有$k$⊆$j$， 在哪里$j$证人${我}_{0,\lambda }$在$\mathrm{中号}$。这个结果首先由作者在附加假设下证明：$V_{\lambda +1}^{\#}$存在；加布·戈德堡注意到这个额外的假设是不必要的。如果还有$\lambda$是一个极限序数并且$\lambda$-DC 坚持$V$，那么模型$\mathrm{中号}$也会满足$\lambda$-DC。

我们证明 ZFC$+$“$\lambda$甚至”$+$${我}_{0,\lambda }$暗示$A^{\#}$存在于每一个$A\epsilon V_{\lambda +1}$，但如果一致，这个理论并不意味着$V_{\lambda +1}^{\#}$存在。