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  1. Patterns of resemblance of order 2.Timothy J. Carlson - 2009 - Annals of Pure and Applied Logic 158 (1-2):90-124.
    We will investigate patterns of resemblance of order 2 over a family of arithmetic structures on the ordinals. In particular, we will show that they determine a computable well ordering under appropriate assumptions.
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  • Subtle cardinals and linear orderings.Harvey M. Friedman - 2000 - Annals of Pure and Applied Logic 107 (1-3):1-34.
    The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen. The concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in 1975 by J. Baumgartner. In this paper we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established by J. Baumgartner. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three (...)
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  • omega ¹-Constructible universe and measurable cardinals.Claude Sureson - 1986 - Annals of Pure and Applied Logic 30 (3):293.
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  • Order types of ordinals in models of set theory.John E. Hutchinson - 1976 - Journal of Symbolic Logic 41 (2):489-502.
    An ordinal in a model of set theory is truly countable if its set of predecessors is countable in the real world. We classify the order types of the sets of truly countable ordinals. Models with indiscernibles and other related results are discussed.
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  • One hundred and two problems in mathematical logic.Harvey Friedman - 1975 - Journal of Symbolic Logic 40 (2):113-129.
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  • Models as Fundamental Entities in Set Theory: A Naturalistic and Practice-based Approach.Carolin Antos - 2022 - Erkenntnis 89 (4):1683-1710.
    This article addresses the question of fundamental entities in set theory. It takes up J. Hamkins’ claim that models of set theory are such fundamental entities and investigates it using the methodology of P. Maddy’s naturalism, Second Philosophy. In accordance with this methodology, I investigate the historical case study of the use of models in the introduction of forcing, compare this case to contemporary practice and give a systematic account of how set-theoretic practice can be said to introduce models as (...)
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  • A new class of order types.James E. Baumgartner - 1976 - Annals of Mathematical Logic 9 (3):187.
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  • Ackermann's set theory equals ZF.William N. Reinhardt - 1970 - Annals of Mathematical Logic 2 (2):189.
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  • Flipping properties: A unifying thread in the theory of large cardinals.F. G. Abramson, L. A. Harrington, E. M. Kleinberg & W. S. Zwicker - 1977 - Annals of Mathematical Logic 12 (1):25.
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  • Some combinatorial problems concerning uncountable cardinals.Thomas J. Jech - 1973 - Annals of Mathematical Logic 5 (3):165.
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  • The number of pairwise non-elementarily-embeddable models.Saharon Shelah - 1989 - Journal of Symbolic Logic 54 (4):1431-1455.
    We get consistency results on I(λ, T 1 , T) under the assumption that D(T) has cardinality $>|T|$ . We get positive results and consistency results on IE(λ, T 1 , T). The interest is model-theoretic, but the content is mostly set-theoretic: in Theorems 1-3, combinatorial; in Theorems 4-7 and 11(2), to prove consistency of counterexamples we concentrate on forcing arguments; and in Theorems 8-10 and 11(1), combinatorics for counterexamples; the rest are discussion and problems. In particular: (A) By Theorems (...)
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  • Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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  • The model< i> N=∪{< i> L_[A]:< i> A countable set of ordinals}.Claude Sureson - 1987 - Annals of Pure and Applied Logic 36 (C):289-313.
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  • A new class of order types.James E. Baumgartner - 1976 - Annals of Mathematical Logic 9 (3):187-222.
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  • Generalized erdoös cardinals and O4.James E. Baumgartner & Fred Galvin - 1978 - Annals of Mathematical Logic 15 (3):289-313.
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  • Σ1-well-founded compactness.Nigel Cutland & Matt Kauffmann - 1980 - Annals of Mathematical Logic 18 (3):271-296.
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  • Instances of dependent choice and the measurability of ℵω + 1.Arthur W. Apter & Menachem Magidor - 1995 - Annals of Pure and Applied Logic 74 (3):203-219.
    Starting from cardinals κ κ is measurable, we construct a model for the theory “ZF + n < ω[DCn] + ω + 1 is a measurable cardinal”. This is the maximum amount of dependent choice consistent with the measurability of ω + 1, and by a theorem of Shelah using p.c.f. theory, is the best result of this sort possible.
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  • A remark on the tree property in a choiceless context.Arthur W. Apter - 2011 - Archive for Mathematical Logic 50 (5-6):585-590.
    We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm ZF} + \neg{\rm AC}_\omega}$$\end{document} + Every successor cardinal is regular + Every limit cardinal (...)
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  • Believing the axioms. I.Penelope Maddy - 1988 - Journal of Symbolic Logic 53 (2):481-511.
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  • The härtig quantifier: A survey.Heinrich Herre, Michał Krynicki, Alexandr Pinus & Jouko Väänänen - 1991 - Journal of Symbolic Logic 56 (4):1153-1183.
    A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Hartig quantifier is, roughly speaking, a first-order language in which the notion of equicardinality is expressible. Thus this language, denoted by LI, is in some sense very natural and has in consequence special interest. Properties of LI are studied in many papers. In [BF, Chapter VI] there is a short survey of some known results about LI. We feel that a more extensive exposition (...)
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  • Core models.A. J. Dodd - 1983 - Journal of Symbolic Logic 48 (1):78-90.
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  • Canonical partition relations.James E. Baumgartner - 1975 - Journal of Symbolic Logic 40 (4):541-554.
    Several canonical partition theorems are obtained, including a simultaneous generalization of Neumer's lemma and the Erdos-Rado theorem. The canonical partition relation for infinite cardinals is completely determined, answering a question of Erdos and Rado. Counterexamples are given showing that in several ways these results cannot be improved.
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  • Logicality and model classes.Juliette Kennedy & Jouko Väänänen - 2021 - Bulletin of Symbolic Logic 27 (4):385-414.
    We ask, when is a property of a model a logical property? According to the so-called Tarski–Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46]. We investigate which characteristics of logics, such as variants of the Löwenheim–Skolem theorem, Completeness theorem, and absoluteness, are relevant from the logicality (...)
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  • Some weak versions of large cardinal axioms.Keith J. Devlin - 1973 - Annals of Mathematical Logic 5 (4):291.
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  • Absolute logics and L∞ω.K. Jon Barwise - 1972 - Annals of Mathematical Logic 4 (3):309-340.
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  • On κ-like structures which embed stationary and closed unbounded subsets.James H. Schmerl - 1976 - Annals of Mathematical Logic 10 (3-4):289-314.
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  • Models of real-valued measurability.Sakae Fuchino, Noam Greenberg & Saharon Shelah - 2006 - Annals of Pure and Applied Logic 142 (1):380-397.
    Solovay’s random-real forcing [R.M. Solovay, Real-valued measurable cardinals, in: Axiomatic Set Theory , Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428] is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovay’s model that do not follow from the existence of real-valued measurability.
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  • Singular cardinals and the pcf theory.Thomas Jech - 1995 - Bulletin of Symbolic Logic 1 (4):408-424.
    §1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals (...)
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  • Some applications of iterated ultrapowers in set theory.Kenneth Kunen - 1970 - Annals of Mathematical Logic 1 (2):179.
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  • Bounds on Scott rank for various nonelementary classes.David Marker - 1990 - Archive for Mathematical Logic 30 (2):73-82.
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  • Abstract logic and set theory. II. large cardinals.Jouko Väänänen - 1982 - Journal of Symbolic Logic 47 (2):335-346.
    The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals.
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  • Strong partition properties for infinite cardinals.E. M. Kleinberg - 1970 - Journal of Symbolic Logic 35 (3):410-428.
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  • The model "N" = [union].Claude Sureson - 1987 - Annals of Pure and Applied Logic 36:289.
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  • On the standard part of nonstandard models of set theory.Menachem Magidor, Saharon Shelah & Jonathan Stavi - 1983 - Journal of Symbolic Logic 48 (1):33-38.
    We characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).
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  • On large cardinals and partition relations.E. M. Kleinberg & R. A. Shore - 1971 - Journal of Symbolic Logic 36 (2):305-308.
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  • The Consistency Strength of $$\aleph{\omega}$$ and $$\aleph_{{\omega}1}$$ Being Rowbottom Cardinals Without the Axiom of Choice.Arthur W. Apter & Peter Koepke - 2006 - Archive for Mathematical Logic 45 (6):721-737.
    We show that for all natural numbers n, the theory “ZF + DC $_{\aleph_n}$ + $\aleph_{\omega}$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $\aleph_{\omega_1}$ is an ω 2-Rowbottom cardinal carrying an ω 2-Rowbottom filter and ω 1 is regular” has the same consistency strength as the theory “ZFC + There exist ω 1 measurable cardinals”. We (...)
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  • On projective ordinals.Alexander S. Kechris - 1974 - Journal of Symbolic Logic 39 (2):269-282.
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  • Classes of Barren Extensions.Natasha Dobrinen & Dan Hathaway - 2021 - Journal of Symbolic Logic 86 (1):178-209.
    Henle, Mathias, and Woodin proved in [21] that, provided that${\omega }{\rightarrow }({\omega })^{{\omega }}$holds in a modelMof ZF, then forcing with$([{\omega }]^{{\omega }},{\subseteq }^*)$overMadds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model$M[\mathcal {U}]$, where$\mathcal {U}$is a Ramsey ultrafilter, with many properties of the original modelM. This begged the question of how important the Ramseyness of$\mathcal (...)
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  • On the consistency of self-referential systems.J. Zimbarg Sobrinho - 1987 - Journal of Symbolic Logic 52 (2):425-436.
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  • The model N = ∪ {L[A]: A countable set of ordinals}.Claude Sureson - 1987 - Annals of Pure and Applied Logic 36:289-313.
    This paper continues the study of covering properties of models closed under countable sequences. In a previous article we focused on C. Chang's Model . Our purpose is now to deal with the model N = ∪ { L [A]: A countable ⊂ Ord}. We study here relations between covering properties, satisfaction of ZF by N , and cardinality of power sets. Under large cardinal assumptions N is strictly included in Chang's Model C , it may thus be interesting to (...)
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  • A saturation property of ideals and weakly compact cardinals.Joji Takahashi - 1986 - Journal of Symbolic Logic 51 (3):513-525.
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  • A Relativization of Axioms of Strong Infinity to ^|^omega;1.Gaisi Takeuti - 1970 - Annals of the Japan Association for Philosophy of Science 3 (5):191-204.
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  • On -like structures which embed stationary and closed unbounded subsets.James H. Schmerl - 1976 - Annals of Mathematical Logic 10 (3):289.
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  • Forcing and reducibilities. III. forcing in fragments of set theory.Piergiorgio Odifreddi - 1983 - Journal of Symbolic Logic 48 (4):1013-1034.
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  • Regressive partitions and borel diagonalization.Akihiro Kanamori - 1989 - Journal of Symbolic Logic 54 (2):540-552.
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  • Rowbottom cardinals and Jonsson cardinals are almost the same.E. M. Kleinberg - 1973 - Journal of Symbolic Logic 38 (3):423-427.
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  • Ordinal definability in the rank hierarchy.John W. Dawson - 1973 - Annals of Mathematical Logic 6 (1):1.
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  • Models of set theory with more real numbers than ordinals.Paul E. Cohen - 1974 - Journal of Symbolic Logic 39 (3):579-583.
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  • Boolean extensions which efface the mahlo property.William Boos - 1974 - Journal of Symbolic Logic 39 (2):254-268.
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  • Absolute logics and L∞ω.K. Jon Barwise - 1972 - Annals of Mathematical Logic 4 (3):309-340.
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