19 found
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  1.  21
    The exact strength of the class forcing theorem.Victoria Gitman, Joel David Hamkins, Peter Holy, Philipp Schlicht & Kameryn J. Williams - 2020 - Journal of Symbolic Logic 85 (3):869-905.
    The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for the (...)
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  2.  20
    What is the theory without power set?Victoria Gitman, Joel David Hamkins & Thomas A. Johnstone - 2016 - Mathematical Logic Quarterly 62 (4-5):391-406.
    We show that the theory, consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there are sets of (...)
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  3.  14
    Virtual large cardinals.Victoria Gitman & Ralf Schindler - 2018 - Annals of Pure and Applied Logic 169 (12):1317-1334.
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  4.  15
    A model of the generic Vopěnka principle in which the ordinals are not Mahlo.Victoria Gitman & Joel David Hamkins - 2019 - Archive for Mathematical Logic 58 (1-2):245-265.
    The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
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  5. A Natural Model of the Multiverse Axioms.Victoria Gitman & Joel David Hamkins - 2010 - Notre Dame Journal of Formal Logic 51 (4):475-484.
    If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins.
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  6.  20
    Ramsey-like cardinals.Victoria Gitman - 2011 - Journal of Symbolic Logic 76 (2):519 - 540.
    One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with (...)
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  7.  20
    Ramsey-like cardinals II.Victoria Gitman & P. D. Welch - 2011 - Journal of Symbolic Logic 76 (2):541-560.
  8. Modern Class Forcing.Carolin Antos & Victoria Gitman - forthcoming - In D. Gabbay M. Fitting (ed.), Research Trends in Contemporary Logic. College Publications.
    We survey recent developments in the theory of class forcing for- malized in the second-order set-theoretic setting.
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  9.  22
    A model of second-order arithmetic satisfying AC but not DC.Sy-David Friedman, Victoria Gitman & Vladimir Kanovei - 2019 - Journal of Mathematical Logic 19 (1):1850013.
    We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. (...)
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  10.  79
    Inner models with large cardinal features usually obtained by forcing.Arthur W. Apter, Victoria Gitman & Joel David Hamkins - 2012 - Archive for Mathematical Logic 51 (3-4):257-283.
    We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ and another in which the least strongly compact cardinal is supercompact. If there is a (...)
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  11.  22
    Ehrenfeucht’s Lemma in Set Theory.Gunter Fuchs, Victoria Gitman & Joel David Hamkins - 2018 - Notre Dame Journal of Formal Logic 59 (3):355-370.
    Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying V=HOD. We show that the lemma fails in the forcing extension of the universe by adding a Cohen real. (...)
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  12.  13
    Easton's theorem for Ramsey and strongly Ramsey cardinals.Brent Cody & Victoria Gitman - 2015 - Annals of Pure and Applied Logic 166 (9):934-952.
  13.  24
    Scott's problem for Proper Scott sets.Victoria Gitman - 2008 - Journal of Symbolic Logic 73 (3):845-860.
    Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω₁. Here, I show that assuming the Proper Forcing Axiom (PFA), every A-proper Scott set is the standard system of a model of PA. I define that a Scott set X is proper if the quotient Boolean algebra X/Fin is a proper partial order and A-proper (...)
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  14.  27
    Indestructibility properties of remarkable cardinals.Yong Cheng & Victoria Gitman - 2015 - Archive for Mathematical Logic 54 (7-8):961-984.
    Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document} is absolute for proper forcing :176–184, 2000). Here, we study the indestructibility properties of remarkable cardinals. We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestructible by all <κ-closed ≤κ-distributive forcing and all two-step iterations of (...)
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  15.  2
    Structural Properties of the Stable Core.Sy-David Friedman, Victoria Gitman & Sandra Müller - 2023 - Journal of Symbolic Logic 88 (3):889-918.
    The stable core, an inner model of the form $\langle L[S],\in, S\rangle $ for a simply definable predicate S, was introduced by the first author in [8], where he showed that V is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname {GCH} $ can fail at all regular cardinals in the stable core, that the stable core can have a discrete (...)
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  16.  9
    Forcing a □(κ)-like principle to hold at a weakly compact cardinal.Brent Cody, Victoria Gitman & Chris Lambie-Hanson - 2021 - Annals of Pure and Applied Logic 172 (7):102960.
  17.  23
    Proper and piecewise proper families of reals.Victoria Gitman - 2009 - Mathematical Logic Quarterly 55 (5):542-550.
    I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ ω1.Here, (...)
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  18.  8
    Incomparable ω 1 ‐like models of set theory.Gunter Fuchs, Victoria Gitman & Joel David Hamkins - 2017 - Mathematical Logic Quarterly 63 (1-2):66-76.
    We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω1‐like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of many ω1‐like models of, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω1‐like model of that does not embed into its own (...)
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  19.  4
    Indestructibility properties of Ramsey and Ramsey-like cardinals.Victoria Gitman & Thomas A. Johnstone - 2022 - Annals of Pure and Applied Logic 173 (6):103106.
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