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28Cichoń’s diagram, regularity properties and $${\varvec{\Delta}^1_3}$$ Δ 3 1 sets of realsArchive for Mathematical Logic 53 (5-6): 695-729. 2014.We study regularity properties related to Cohen, random, Laver, Miller and Sacks forcing, for sets of real numbers on the Δ31\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Delta}^1_3}$$\end{document} level of the projective hieararchy. For Δ21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{w…Read more
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28Large cardinals and lightface definable well-orders, without the gchJournal of Symbolic Logic 80 (1): 251-284. 2015.
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27Review: Donald A. Martin, A. S. Kechris, D. A. Martin, Y. N. Moschovakis, The Largest Countable this, that, and the other; Alexander S. Kechris, Donald A. Martin, Robert M. Solovay, Introduction to $Q$-Theory; Steve Jackson, A. S. Kechris, D. A. Martin, J. R. Steel, AD and the Projective Ordinals (review)Journal of Symbolic Logic 57 (1): 262-264. 1992.
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26Baumgartnerʼs conjecture and bounded forcing axiomsAnnals of Pure and Applied Logic 164 (12): 1178-1186. 2013.
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25Potential isomorphism of elementary substructures of a strictly stable homogeneous modelJournal of Symbolic Logic 76 (3). 2011.The results herein form part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability (in the sense of [1]) of the potential isomorphism problem for this class of submodels. We restrict ourselves to locally saturated submodels of the monster model m of some power π. We assume that in Gödel's constructible universe , π is a regular cardinal at least the successor of the first cardinal in which is stabl…Read more
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25The stable coreBulletin of Symbolic Logic 18 (2): 261-267. 2012.Vopenka [2] proved long ago that every set of ordinals is set-generic over HOD, Gödel's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over, and indeed over the even smaller inner model $\mathbb{S}=$, where S is the Stability predicate. We refer to the inner model $\mathbb{S}$ as the Stable Core of V. The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible to add reals w…Read more
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25On Absoluteness of Categoricity in Abstract Elementary ClassesNotre Dame Journal of Formal Logic 52 (4): 395-402. 2011.Shelah has shown that $\aleph_1$-categoricity for Abstract Elementary Classes (AECs) is not absolute in the following sense: There is an example $K$ of an AEC (which is actually axiomatizable in the logic $L(Q)$) such that if $2^{\aleph_0}
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24Beller A., Jensen R., and Welch P.. Coding the universe. London Mathematical Society lecture note series, no. 47. Cambridge University Press, Cambridge etc. 1982, 353 pp (review)Journal of Symbolic Logic 50 (4): 1081-1081. 1985.
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24The tree property at the double successor of a singular cardinal with a larger gapAnnals of Pure and Applied Logic 169 (6): 548-564. 2018.
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23Generic coding with help and amalgamation failureJournal of Symbolic Logic 86 (4): 1385-1395. 2021.We show that if M is a countable transitive model of $\text {ZF}$ and if $a,b$ are reals not in M, then there is a G generic over M such that $b \in L[a,G]$. We then present several applications such as the following: if J is any countable transitive model of $\text {ZFC}$ and $M \not \subseteq J$ is another countable transitive model of $\text {ZFC}$ of the same ordinal height $\alpha $, then there is a forcing extension N of J such that $M \cup N$ is not included in any transitive model of $\t…Read more
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23A simpler proof of Jensen's coding theoremAnnals of Pure and Applied Logic 70 (1): 1-16. 1994.Jensen's remarkable Coding Theorem asserts that the universe can be included in L[R] for some real R, via class forcing. The purpose of this article is to present a simpler proof of Jensen's theorem, obtained by implementing some changes first developed for the theory of Strong Coding. In particular, our proof avoids the split into cases, according to whether or not 0# exists in the ground model
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23The Nonabsoluteness of Model Existence in Uncountable Cardinals for $L{omega{1},omega}$Notre Dame Journal of Formal Logic 54 (2): 137-151. 2013.For sentences $\phi$ of $L_{\omega_{1},\omega}$, we investigate the question of absoluteness of $\phi$ having models in uncountable cardinalities. We first observe that having a model in $\aleph_{1}$ is an absolute property, but having a model in $\aleph_{2}$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis context and provide sentences for any $\alpha\in\omega_{1}\setminus\{0,1,\omega\}$ for which the existence of a model…Read more
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22Classification theory and 0#Journal of Symbolic Logic 68 (2): 580-588. 2003.We characterize the classifiability of a countable first-order theory T in terms of the solvability of the potential-isomorphism problem for models of T.
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22Easton’s theorem and large cardinalsAnnals of Pure and Applied Logic 154 (3): 191-208. 2008.The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α
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22Collapsing the cardinals of HODJournal of Mathematical Logic 15 (2): 1550007. 2015.Assuming that GCH holds and [Formula: see text] is [Formula: see text]-supercompact, we construct a generic extension [Formula: see text] of [Formula: see text] in which [Formula: see text] remains strongly inaccessible and [Formula: see text] for every infinite cardinal [Formula: see text]. In particular the rank-initial segment [Formula: see text] is a model of ZFC in which [Formula: see text] for every infinite cardinal [Formula: see text].
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22Cardinal characteristics at κ in a small u ( κ ) modelAnnals of Pure and Applied Logic 168 (1): 37-49. 2017.
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21Large cardinals and definable well-orders, without the GCHAnnals of Pure and Applied Logic 166 (3): 306-324. 2015.
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21The eightfold wayJournal of Symbolic Logic 83 (1): 349-371. 2018.Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniqu…Read more
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20The tree property at א ω+2Journal of Symbolic Logic 76 (2). 2011.Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension א ω is a strong limit cardinal and א ω+2 has the tree property. This improves a result of Matthew Foreman (see [2])
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20Evidence for Set-Theoretic Truth and the Hyperuniverse ProgrammeIn Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 75-107. 2018.I discuss three potential sources of evidence for truth in set theory, coming from set theory’s roles as a branch of mathematics and as a foundation for mathematics as well as from the intrinsic maximality feature of the set concept. I predict that new non first-order axioms will be discovered for which there is evidence of all three types, and that these axioms will have significant first-order consequences which will be regarded as true statements of set theory. The bulk of the paper is concer…Read more
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20To show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a union L=⋃Bulletin of Symbolic Logic 3 (4): 453-468. 1997.We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the traditional Lα -or Jα-seque…Read more
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20On strong forms of reflection in set theoryMathematical Logic Quarterly 62 (1-2): 52-58. 2016.In this paper we review the most common forms of reflection and introduce a new form which we call sharp‐generated reflection. We argue that sharp‐generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp‐maximality with the corresponding hypothesis. The statement is an analogue of the (Inner Model Hypothesis, introduced in ) which is compatible with the existence o…Read more
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19Jensen's $Sigma^ast$ Theory and the Combinatorial Content of $V = L$Journal of Symbolic Logic 59 (3): 1096-1104. 1994.
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19Failures of the silver dichotomy in the generalized baire spaceJournal of Symbolic Logic 80 (2): 661-670. 2015.We prove results that falsify Silver’s dichotomy for Borel equivalence relations on the generalized Baire space under the assumptionV=L.