Reinhardt cardinals and non-definability
Reinhardt cardinals and non-definability
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莱因哈特基数和不可定义性
DOI：
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Farmer Schlutzenberg
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文献类型：
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作者：
Farmer Schlutzenberg
Work in $\mathsf{ZF}$ or $\mathsf{ZF}_2$ (second order $\mathsf{ZF}$), as appropriate. Recall that a Reinhardt cardinal is the critical point of a (non-trivial) elementary embedding $j:V\rightarrow V$. Beyond these, one has super-Reinhardt, total Reinhardt and Berkeley cardinals. We prove the following results. Let $X$ be a set and $A$ a class. Then (i) if there is a Reinhardt cardinal then $V\neq\mathrm{HOD}(X)$, and (ii) if $V$ is total Reinhardt or there is a Berkeley cardinal then $V\neq\mathrm{HOD}_A(X)$. Let $\delta$ be a limit ordinal and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Then (i) $j$ is not definable from parameters over $V_\delta$, and (ii) there is $n<\omega$ such that the $n^{\mathrm{th}}$ iterate $j^n=j(j(\ldots (j))):V_\delta\to V_\delta$ is fully elementary; in fact, $j^n:(V_\delta,A)\to(V_\delta,j^n(A))$ is fully elementary for all $A\subseteq V_\delta$. Let $\delta$ be any ordinal and $j:V_{\delta+1}\to V_{\delta+1}$ be fully elementary. Then $j$ is not definable over $V_{\delta+1}$ from parameters in $V_\delta$. Suppose $V=L(V_\delta)$ and $\mathrm{cof}(\delta)>\omega$. Then there is no $\Sigma_1$-elementary $j:V_\delta\to V_\delta$. Let $G$ be $(V,\mathbb{P})$-generic for some $\mathbb{P}\in V$. Then (i) if $V[G]$ has a super-Reinhardt cardinal, then $V$ has a super-Reinhardt cardinal; (ii) if $\mathbb{P}\in V_\delta$ and $V[G]\models$``$V_\delta^{V[G]}$ is total Reinhardt'' then $V\models$``$V_\delta$ is total Reinhardt''; and (iii) if $V[G]$ has a set of ordinals which is not in $V$, then $V[G]$ has no elementary $j:V[G]\to M\subseteq V$. We also develop the theory of ultrapowers by extenders under $\mathsf{ZF}$, and show that if there is a proper class of L\"owenheim-Skolem cardinals, then being the critical point of an elementary $j:V\to M$ (with $M$ transitive) is first-order.
Work in $\mathsf{ZF}$ or $\mathsf{ZF}_2$ (second order $\mathsf{ZF}$), as appropriate. Recall that a Reinhardt cardinal is the critical point of a (non-trivial) elementary embedding $j:V\rightarrow V$. Beyond these, one has super-Reinhardt, total Reinhardt and Berkeley cardinals. We prove the following results. Let $X$ be a set and $A$ a class. Then (i) if there is a Reinhardt cardinal then $V\neq\mathrm{HOD}(X)$, and (ii) if $V$ is total Reinhardt or there is a Berkeley cardinal then $V\neq\mathrm{HOD}_A(X)$. Let $\delta$ be a limit ordinal and $j:V_\delta\to V_\delta$ be $\Sigma_1$-elementary. Then (i) $j$ is not definable from parameters over $V_\delta$, and (ii) there is $n<\omega$ such that the $n^{\mathrm{th}}$ iterate $j^n=j(j(\ldots (j))):V_\delta\to V_\delta$ is fully elementary; in fact, $j^n:(V_\delta,A)\to(V_\delta,j^n(A))$ is fully elementary for all $A\subseteq V_\delta$. Let $\delta$ be any ordinal and $j:V_{\delta+1}\to V_{\delta+1}$ be fully elementary. Then $j$ is not definable over $V_{\delta+1}$ from parameters in $V_\delta$. Suppose $V=L(V_\delta)$ and $\mathrm{cof}(\delta)>\omega$. Then there is no $\Sigma_1$-elementary $j:V_\delta\to V_\delta$. Let $G$ be $(V,\mathbb{P})$-generic for some $\mathbb{P}\in V$. Then (i) if $V[G]$ has a super-Reinhardt cardinal, then $V$ has a super-Reinhardt cardinal; (ii) if $\mathbb{P}\in V_\delta$ and $V[G]\models$``$V_\delta^{V[G]}$ is total Reinhardt'' then $V\models$``$V_\delta$ is total Reinhardt''; and (iii) if $V[G]$ has a set of ordinals which is not in $V$, then $V[G]$ has no elementary $j:V[G]\to M\subseteq V$. We also develop the theory of ultrapowers by extenders under $\mathsf{ZF}$, and show that if there is a proper class of L\"owenheim-Skolem cardinals, then being the critical point of an elementary $j:V\to M$ (with $M$ transitive) is first-order.
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DOI：
10.1017/bsl.2019.28
发表时间：
2019
期刊：
The Bulletin of Symbolic Logic
影响因子：
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作者：
BAGARIA, JOAN;KOELLNER, PETER;WOODIN, W. HUGH
通讯作者：
WOODIN, W. HUGH