论文标题
从$ v_ {λ+2} $纳入$ v_ {λ+2} $的ZF的一致性与基本嵌入$
On the consistency of ZF with an elementary embedding from $V_{λ+2}$ into $V_{λ+2}$
论文作者
论文摘要
根据肯尼斯·库嫩（Kenneth Kunen）的定理，根据ZFC，没有序数$λ$和非平凡的基本嵌入$ j：v_ {λ+2} \ to v_ {λ+2} $。他的证明依赖于选择的公理（AC），并且没有发现ZF的证据。 $ i_ {0，λ} $是W. Hugh Woodin引入的断言，$λ$是序数，并且有一个基本嵌入$ j：l（v_ {λ+1}）\ to l（v_ {λ+1}）$，带有关键点$ {<λ} $。 $ i_0 $断言$ i_ {0，λ} $对于某些$λ$保留。 Axiom $ I_0 $是最强大的大型红衣主教之一，与AC不一致。通常对整个Universe $ V $中的ZFC进行研究（在这种情况下，$λ$必须是极限序列），但我们仅假定ZF。 我们证明，假设ZF + $ i_ {0，λ} $ +“ $λ$是一个偶数”，那就是有一个适当的类传递内部模型$ m $，其中包含$ v_ {λ + 1} $，并且满足Zf + $ i_ {0，λ} $ +”，有一个基本的嵌入$ K： v_ {λ+2} $“;实际上，我们将有$ k \ subseteq j $，其中$ j $ niventle $ i_ {0，λ} $ in $ m $。作者首先证明了该结果的额外假设，即存在$ v_ {λ+1}^\＃$ $；加布·戈德堡（Gabe Goldberg）注意到，这种额外的假设是不必要的。如果$λ$也是限制序列，而$λ$ -DC在$ v $中持有，那么$ m $也将满足$λ$ -DC。 我们表明ZFC +“ $λ$偶数” + $ i_ {0，λ} $暗示$ a^\＃$都存在于v_ {λ + 1} $中的每个$ a \ in \ a \ y \ y \＃$，但是如果一致，则该理论并不意味着$ v_ {λ + 1}}^\＃$存在。
According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $λ$ and non-trivial elementary embedding $j:V_{λ+2}\to V_{λ+2}$. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone has been discovered. $I_{0,λ}$ is the assertion, introduced by W. Hugh Woodin, that $λ$ is an ordinal and there is an elementary embedding $j:L(V_{λ+1})\to L(V_{λ+1})$ with critical point ${<λ}$. And $I_0$ asserts that $I_{0,λ}$ holds for some $λ$. 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 $λ$ must be a limit ordinal), but we assume only ZF. We prove, assuming ZF + $I_{0,λ}$ + "$λ$ is an even ordinal", that there is a proper class transitive inner model $M$ containing $V_{λ+1}$ and satisfying ZF + $I_{0,λ}$ + "there is an elementary embedding $k:V_{λ+2}\to V_{λ+2}$"; in fact we will have $k\subseteq j$, where $j$ witnesses $I_{0,λ}$ in $M$. This result was first proved by the author under the added assumption that $V_{λ+1}^\#$ exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also $λ$ is a limit ordinal and $λ$-DC holds in $V$, then the model $M$ will also satisfy $λ$-DC. We show that ZFC + "$λ$ is even" + $I_{0,λ}$ implies $A^\#$ exists for every $A\in V_{λ+1}$, but if consistent, this theory does not imply $V_{λ+1}^\#$ exists.
