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Farmer Schlutzenberg - One of the best experts on this subject based on the ideXlab platform.
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On the consistency of ZF with an Elementary Embedding from $V_{\lambda+2}$ into $V_{\lambda+2}$
arXiv: Logic, 2020Co-Authors: Farmer SchlutzenbergAbstract:Recall that $I_{0,\lambda}$ is the assertion that $\lambda$ is a limit ordinal and there is an Elementary Embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point ${
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on the consistency of zf with an Elementary Embedding from v_ lambda 2 into v_ lambda 2
arXiv: Logic, 2020Co-Authors: Farmer SchlutzenbergAbstract:Recall that $I_{0,\lambda}$ is the assertion that $\lambda$ is a limit ordinal and there is an Elementary Embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point ${<\lambda}$. This hypothesis is usually studied assuming ZFC holds in the full universe $V$, but we assume only ZF. We show, assuming ZF+$I_{0,\lambda}$, that there is a proper class transitive inner model $M$ containing $V_{\lambda+1}$ and modelling the theory \[ \mathrm{ZF}+I_{0,\lambda}+\text{"there is an Elementary Embedding }j:V_{\lambda+2}\to V_{\lambda+2}\text{"}.\] By employing the results of the papers \emph{Periodicity in the cumulative hierarchy} and \emph{Even ordinals and the Kunen inconsistency}, we also show that this generalizes to all even ordinals $\lambda$. In the case that $\lambda$ is a limit and $\lambda$-DC holds in $V$, then the model $M$ constructed also satisfies $\lambda$-DC. We also show that if ZFC+$I_{0,\lambda}$ is consistent, then it does not imply the existence of $V_{\lambda+1}^\#$. Likewise, if ZF+"$\lambda$ is an even ordinal and $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ is Elementary with critical point ${<\lambda}$" is consistent, then it does not imply the existence of $V_{\lambda+1}^\#$. We show that, however, this theory does imply that $A^\#$ exists for every $A\in V_{\lambda+1}$. We also make some further obserations on $L(V_{\lambda+1})$ under such hypotheses.
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reinhardt cardinals and non definability
arXiv: Logic, 2020Co-Authors: Farmer SchlutzenbergAbstract: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 Lowenheim-Skolem cardinals, then being the critical point of an Elementary $j:V\to M$ (with $M$ transitive) is first-order.
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reinhardt cardinals and iterates of v
arXiv: Logic, 2020Co-Authors: Farmer SchlutzenbergAbstract:Assume ZF($j$) and there is a Reinhardt cardinal, as witnessed by the Elementary Embedding $j:V\to V$. We investigate the linear iterates $(N_{\alpha},j_{\alpha})$ of $(V,j)$, and their relationship to $(V,j)$, forcing and definability, including that for each infinite ordinal $\alpha$, every set is set-generic over $N_{\alpha}$, but $N_{\alpha}$ is not a set-ground. Assume second order ZF. We prove that the existence of super Reinhardt cardinals and total Reinhardt cardinals is not affected by small forcing. And if $V[G]$ has a set of ordinals which is not in $V$, then $V[G]$ has no Elementary Embedding $j:V[G]\to M\subseteq V$ (even allowing $M$ to be illfounded).
Schlutzenberg Farmer - One of the best experts on this subject based on the ideXlab platform.
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Periodicity in the cumulative hierarchy
2020Co-Authors: Goldberg Gabriel, Schlutzenberg FarmerAbstract:We investigate the structure of rank-into-rank Elementary Embeddings incompatible with the Axiom of Choice. Assuming ZF set theory and the existence of a (non-trivial) Elementary Embedding \[ j:V_{\alpha+1}\to V_{\alpha+1}\] of a rank initial segment of the universe $V$ into itself, we show that the structure of $V_\alpha$ is fundamentally different to that of $V_{\alpha+1}$. (Here $\alpha$ may be either a limit or a successor ordinal.) We show that $j$ is definable from parameters over $V_{\alpha+1}$ iff $\alpha+1$ is an odd ordinal. Moreover, if $\alpha+1$ is odd then $j$ is definable over $V_{\alpha+1}$ from its restriction $j\upharpoonright V_{\alpha}$, and uniformly so. This parameter is optimal in that $j$ is not definable from any parameter in $V_\alpha$. Further, we also show that $\Sigma_1$-Elementary Embeddings $j:V_\lambda\to V_\lambda$ are non-definable for all limit $\lambda$. It is moreover known that if there is a Reinhardt cardinal, then for all sufficiently large ordinals $\alpha$, there is an Elementary $j:V_\alpha\to V_\alpha$, and therefore the cumulative hierarchy is eventually \emph{periodic} (with period 2).Comment: 31 page
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Periodicity in the cumulative hierarchy
2020Co-Authors: Goldberg Gabriel, Schlutzenberg FarmerAbstract:We investigate the structure of rank-to-rank Elementary Embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels $V_\alpha$ of the cumulative hierarchy are defined via iterated application of the power set operation, starting from $V_0=\emptyset$, and taking unions at limit stages. Assuming that $j:V_{\alpha+1}\to V_{\alpha+1}$ is a (non-trivial) Elementary Embedding, we show that the structure of $V_\alpha$ is fundamentally different to that of $V_{\alpha+1}$. We show that $j$ is definable from parameters over $V_{\alpha+1}$ iff $\alpha+1$ is an odd ordinal. Moreover, if $\alpha+1$ is odd then $j$ is definable over $V_{\alpha+1}$ from the parameter $j`` V_{\alpha}=\{j(x)\bigm|x\in V_\alpha\}$, and uniformly so. This parameter is optimal in that $j$ is not definable from any parameter which is an element of $V_\alpha$. In the case that $\alpha=\beta+1$, we also give a characterization of such $j$ in terms of ultrapower maps via certain ultrafilters. Assuming $\lambda$ is a limit ordinal, we prove that if $j:V_\lambda\to V_\lambda$ is $\Sigma_1$-Elementary, then $j$ is not definable over $V_\lambda$ from parameters, and if $\beta
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Reinhardt cardinals and iterates of V
2020Co-Authors: Schlutzenberg FarmerAbstract:Assume ZF($j$) and there is a Reinhardt cardinal, as witnessed by the Elementary Embedding $j:V\to V$. We investigate the linear iterates $(N_{\alpha},j_{\alpha})$ of $(V,j)$, and their relationship to $(V,j)$, forcing and definability, including that for each infinite ordinal $\alpha$, every set is set-generic over $N_{\alpha}$, but $N_{\alpha}$ is not a set-ground. Assume second order ZF. We prove that the existence of super Reinhardt cardinals and total Reinhardt cardinals is not affected by small forcing. And if $V[G]$ has a set of ordinals which is not in $V$, then $V[G]$ has no Elementary Embedding $j:V[G]\to M\subseteq V$ (even allowing $M$ to be illfounded).Comment: 23 pages; v1 split into 3 docs: this (v4), 2006.01103, 2006.10574. See v2 comments. Added citations to 1106.1951: in introduction, regarding generalizations of Suzuki 99; in Footnote 3 (p.2) and Remark 3.20 (Theorem 35 of 1106.1951 already establishes a fact discussed at those points); and in the introduction to section 2 (Corollary 34 of 1106.1951 is related). Added Remark 4.7. Other minor edit
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Extenders under ZF and constructibility of rank-to-rank Embeddings
2020Co-Authors: Schlutzenberg FarmerAbstract:Assume ZF (without the Axiom of Choice). Let $j:V_\delta\to V_\delta$ be a non-trivial $\Sigma_1$-Elementary Embedding, where $\delta$ is a limit ordinal. We prove some basic restrictions on the constructibility of $j$ from $V_\delta$; in particular, if $j\in L(V_\delta)$ then $\delta$ has uncountable cofinality. We show that, however, assuming an $I_3$-Embedding, with the appropriate $\delta,j$, it is possible to have $j\in L(V_\delta)$. Assuming Dependent Choice and that $\delta$ has countable cofinality (but not assuming $V=L(V_\delta)$), and $j$ is as above, we show that the collection of such Embeddings is of high complexity, and that there are "perfectly many" such Embeddings. We also show that a ZF theorem of Suzuki, that no Elementary $j:V\to V$ is definable from parameters, actually follows from a theory weaker than ZF. The main results rely on a development of extenders under ZF, which we also give.Comment: 28 pages. This version: Changed title, expanded introduction, other small edits. Note: the author has split arXiv:2002.01215v1 into separate components, and this paper constitutes one of those components, and hence draws heavily on those note
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Extenders under ZF and constructibility of rank-to-rank Embeddings
2020Co-Authors: Schlutzenberg FarmerAbstract:Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_\delta$ be a non-trivial $\in$-cofinal $\Sigma_1$-Elementary Embedding, where $\varepsilon,\delta$ are limit ordinals. We prove some restrictions on the constructibility of $j$ from $V_\delta$, mostly focusing on the case $\varepsilon=\delta$. In particular, if $\varepsilon=\delta$ and $j\in L(V_\delta)$ then $\delta$ has cofinality $\omega$. However, assuming ZFC+I$_3$, with the appropriate $\varepsilon=\delta$, one can force to get such $j\in L(V^{V[G]}_\delta)$. Assuming Dependent Choice and that $\delta$ has cofinality $\omega$ (but not assuming $V=L(V_\delta)$), and $j:V_\delta\to V_\delta$ is $\Sigma_1$-Elementary, we show that there are "perfectly many" such $j$, with none being "isolated". Assuming a proper class of weak Lowenheim-Skolem cardinals, we also give a first-order characterization of critical points of Embeddings $j:V\to M$ with $M$ transitive. The main results rely on a development of extenders under ZF (which is most useful given such wLS cardinals).Comment: 32 pages. This version: Extended some results (6.9, 7.1, 7.6), and modified introduction accordingly. Corrected typo in abstract which asserted a key theorem falsely. Added URL links to bibliography. arXiv admin note: text overlap with arXiv:2002.0121
Goldberg Gabriel - One of the best experts on this subject based on the ideXlab platform.
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Even ordinals and the Kunen inconsistency
2021Co-Authors: Goldberg GabrielAbstract:This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom. The third and final part of the paper examines the consistency strength of choiceless large cardinals, including a proof that assuming DC, the existence of an Elementary Embedding from $V_{\lambda+3}$ to $V_{\lambda+3}$ implies the consistency of ZFC + $I_0$. By a recent result of Schlutzenberg, an Elementary Embedding from $V_{\lambda+2}$ to $V_{\lambda+2}$ does not suffice.Comment: 60 page
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Measurable cardinals and choiceless axioms
2021Co-Authors: Goldberg GabrielAbstract:We prove that if there is an Elementary Embedding from the universe to itself, then there is a proper class of measurable successor cardinals.Comment: 17 pages. Fills a gap in the proof of the filter extension propert
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Reinhardt cardinals in inner models
2021Co-Authors: Goldberg GabrielAbstract:A cardinal is weakly Reinhardt if it is the critical point of an Elementary Embedding from the universe of sets into a model that contains the double powerset of every ordinal. This note establishes the equiconsistency of a proper class of weakly Reinhardt cardinals with a proper class of Reinhardt cardinals in the context of second-order set theory without the Axiom of Choice.Comment: 4 page
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Strongly compact cardinals and ordinal definability
2021Co-Authors: Goldberg GabrielAbstract:This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no Elementary Embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of Elementary Embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable omega-Jonsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD.Comment: 16 page
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Some combinatorial properties of Ultimate L and V
2020Co-Authors: Goldberg GabrielAbstract:This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality is strongly compact, making progress on a question of Boney, Unger, and Brooke-Taylor. We show that if there is a proper class of strongly compact cardinals then there is no nontrivial cardinal preserving Elementary Embedding from the universe of sets into an inner model, answering a question of Caicedo granting large cardinals. Finally, we show that if \(\kappa\) is strongly compact, then \(V\) is a set forcing extension of the inner model \(\kappa\text{-HOD}\) consisting of sets that are hereditarily ordinal definable from a \(\kappa\)-complete ultrafilter over an ordinal; \(\kappa\text{-HOD}\) seems to be the first nontrivial example of a ground of \(V\) whose definition does not involve forcing.Comment: 33 page
André Simon - One of the best experts on this subject based on the ideXlab platform.
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Elementary SUBGROUPS OF VIRTUALLY FREE GROUPS
HAL CCSD, 2019Co-Authors: André SimonAbstract:19 pages.We give a description of Elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that Elementary subgroups of finitely generated free groups are free factors. Moreover, we give an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $\exists\forall\exists$-Elementary. We also prove that every Elementary Embedding of an equationally noetherian group into itself is an automorphism
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Elementary subgroups of virtually free groups
2019Co-Authors: André SimonAbstract:We give a description of Elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that Elementary subgroups of finitely generated free groups are free factors. Moreover, we give an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $\exists\forall\exists$-Elementary. We also prove that every Elementary Embedding of an equationally noetherian group into itself is an automorphism.Comment: 19 page
Paul Corazza - One of the best experts on this subject based on the ideXlab platform.
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The spectrum of Elementary Embeddings j : V
2015Co-Authors: Paul CorazzaAbstract:Abstract. In 1970, K. Kunen, working in the context of Kelley-Morse set the-ory, showed that the existence of a nontrivial Elementary Embedding j: V → V is inconsistent. In this paper, we give a finer analysis of the implications of his result for Embeddings V → V relative to models of ZFC. We do this by working in the extended language {∈, j}, using as axioms all the usual axioms of ZFC (for ∈-formulas), along with an axiom schema that asserts that j is a nontrivial Elementary Embedding. Without additional axiomatic assumptions on j, we show that that the resulting theory (denoted ZFC + BTEE) is weaker than an ω-Erdös cardinal, but stronger than n-ineffables. We show that natu-ral models of ZFC + BTEE give rise to Schindler’s remarkable cardinals. The approach to inconsistency from ZFC + BTEE forks into two paths: extensions of ZFC + BTEE + Cofinal Axiom and ZFC + BTEE + ¬Cofinal Axiom, where Cofinal Axiom asserts that the critical sequence κ, j(κ), j2(κ),... is cofinal in the ordinals. We describe near-minimal inconsistent extensions of each of these the-ories. The path toward inconsistency from ZFC + BTEE + ¬Cofinal Axiom is paved with a sequence of theories of increasing large cardinal strength. Indeed, the extensions of the theory ZFC+“j is a nontrivial Elementary Embedding ” form a hierarchy of axioms, ranging in strength from Con(ZFC) to the existence of a cardinal that is super-n-huge for every n, to inconsistency. This hierarchy is parallel to the usual hierarchy of large cardinal axioms, and can be used in the same way. We also isolate several intermediate-strength axioms which, when added to ZFC + BTEE, produce theories having strengths in the vicinity of a measurable cardinal of high Mitchell order, a strong cardinal, ω Woodin cardi-nals, and n-huge cardinals. We also determine precisely which combinations of axioms, of the for