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10Structural Properties of the Stable CoreJournal of Symbolic Logic 88 (3): 889-918. 2023.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 proper class of measurable cardinals, but t…Read more
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18Universally Baire sets and definable well-orderings of the realsJournal of Symbolic Logic 68 (4): 1065-1081. 2003.Let n ≥ 3 be an integer. We show that it is consistent that every σ1n-set of reals is universally Baire yet there is a projective well-ordering of the reals. The proof uses “David’s trick” in the presence of inner models with strong cardinals.
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47Some recent developments in higher recursion theoryJournal of Symbolic Logic 48 (3): 629-642. 1983.In recent years higher recursion theory has experienced a deep interaction with other areas of logic, particularly set theory (fine structure, forcing, and combinatorics) and infinitary model theory. In this paper we wish to illustrate this interaction by surveying the progress that has been made in two areas: the global theory of the κ-degrees and the study of closure ordinals
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42Independence of higher Kurepa hypothesesArchive for Mathematical Logic 51 (5-6): 621-633. 2012.We study the Generalized Kurepa hypothesis introduced by Chang. We show that relative to the existence of an inaccessible cardinal the Gap-n-Kurepa hypothesis does not follow from the Gap-m-Kurepa hypothesis for m different from n. The use of an inaccessible is necessary for this result.
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31HC of an admissible setJournal of Symbolic Logic 44 (1): 95-102. 1979.If A is an admissible set, let HC(A) = {x∣ x ∈ A and x is hereditarily countable in A}. Then HC(A) is admissible. Corollaries are drawn characterizing the "real parts" of admissible sets and the analytical consequences of admissible set theory
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18Handbook of mathematical logic, edited by Barwise Jon with the cooperation of Keisler H. J., Kunen K., Moschovakis Y. N., and Troelstra A. S., Studies in logic and the foundations of mathematics, vol. 90, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1978 , xi + 1165 pp (review)Journal of Symbolic Logic 49 (3): 975-980. 1984.
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29Genericity and large cardinalsJournal of Mathematical Logic 5 (02): 149-166. 2005.We lift Jensen's coding method into the context of Woodin cardinals. By a theorem of Woodin, any real which preserves a "strong witness" to Woodinness is set-generic. We show however that there are class-generic reals which are not set-generic but preserve Woodinness, using "weak witnesses".
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14Fragments of Kripke–Platek set theory and the metamathematics of $$\alpha $$ α -recursion theoryArchive for Mathematical Logic 55 (7-8): 899-924. 2016.The foundation scheme in set theory asserts that every nonempty class has an ∈\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\in $$\end{document}-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and α\documentclass[12pt]{minimal} \usepackage{amsmath}…Read more
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Δ1-definabilityAnnals of Pure and Applied Logic 89 (1): 93-99. 1997.We isolate a condition on a class A of ordinals sufficient to Δ1-code it by a real in a class-generic extension of L. We then apply this condition to show that the class of ordinals of L-cofinality ω is Δ1 in a real of L-degree strictly below O#.
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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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36A model of second-order arithmetic satisfying AC but not DCJournal of Mathematical Logic 19 (1): 1850013. 2019.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. G. Kanovei, On…Read more
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63A guide to "coding the universe" by Beller, Jensen, WelchJournal of Symbolic Logic 50 (4): 1002-1019. 1985.
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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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33□ On the singular cardinalsJournal of Symbolic Logic 73 (4): 1307-1314. 2008.We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen. "Square on singular cardinals"
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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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51Large cardinals need not be large in HODAnnals of Pure and Applied Logic 166 (11): 1186-1198. 2015.
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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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46Meeting of the association for symbolic logic: New York 1979Journal of Symbolic Logic 46 (2): 427-434. 1981.