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  1. Perception, Intuition, and Reliability.Kai Hauser & Tahsİn Öner - 2018 - Theoria 84 (1):23-59.
    The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...)
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  • The long extender algebra.Ralf Schindler - 2018 - Archive for Mathematical Logic 57 (1-2):73-82.
    Generalizing Woodin’s extender algebra, cf. e.g. Steel Handbook of set theory, Springer, Berlin, 2010), we isolate the long extender algebra as a general version of Bukowský’s forcing, cf. Bukovský, in the presence of a supercompact cardinal.
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  • The HOD Hypothesis and a supercompact cardinal.Yong Cheng - 2017 - Mathematical Logic Quarterly 63 (5):462-472.
    In this paper, we prove that: if κ is supercompact and the math formula Hypothesis holds, then there is a proper class of regular cardinals in math formula which are measurable in math formula. Woodin also proved this result independently [11]. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the math formula Hypothesis and supercompact cardinals, large cardinals in math formula are reflected to be large cardinals in math formula in a (...)
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  • Axiom I 0 and higher degree theory.Xianghui Shi - 2015 - Journal of Symbolic Logic 80 (3):970-1021.
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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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  • Large cardinals need not be large in HOD.Yong Cheng, Sy-David Friedman & Joel David Hamkins - 2015 - Annals of Pure and Applied Logic 166 (11):1186-1198.
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  • How Woodin changed his mind: new thoughts on the Continuum Hypothesis.Colin J. Rittberg - 2015 - Archive for History of Exact Sciences 69 (2):125-151.
    The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a very different (...)
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  • Boolean-Valued Second-Order Logic.Daisuke Ikegami & Jouko Väänänen - 2015 - Notre Dame Journal of Formal Logic 56 (1):167-190.
    In so-called full second-order logic, the second-order variables range over all subsets and relations of the domain in question. In so-called Henkin second-order logic, every model is endowed with a set of subsets and relations which will serve as the range of the second-order variables. In our Boolean-valued second-order logic, the second-order variables range over all Boolean-valued subsets and relations on the domain. We show that under large cardinal assumptions Boolean-valued second-order logic is more robust than full second-order logic. Its (...)
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  • Intuition and Its Object.Kai Hauser - 2015 - Axiomathes 25 (3):253-281.
    The view that mathematics deals with ideal objects to which we have epistemic access by a kind of perception has troubled many thinkers. Using ideas from Husserl’s phenomenology, I will take a different look at these matters. The upshot of this approach is that there are non-material objects and that they can be recognized in a process very closely related to sense perception. In fact, the perception of physical objects may be regarded as a special case of this more universal (...)
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  • Large cardinals at the brink.W. Hugh Woodin - 2024 - Annals of Pure and Applied Logic 175 (1):103328.
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  • Measurable cardinals and choiceless axioms.Gabriel Goldberg - forthcoming - Annals of Pure and Applied Logic.
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  • The relative strengths of fragments of Martin's axiom.Joan Bagaria - 2024 - Annals of Pure and Applied Logic 175 (1):103330.
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  • More on the Preservation of Large Cardinals Under Class Forcing.Joan Bagaria & Alejandro Poveda - 2023 - Journal of Symbolic Logic 88 (1):290-323.
    We prove two general results about the preservation of extendible and $C^{(n)}$ -extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka’s Principle and $C^{(n)}$ -extendible cardinals under Jensen’s iteration for forcing the GCH [17], previously obtained in [8, 27], respectively. We prove that $C^{(n)}$ -extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Delta _2$ -definable behaviour of the power-set function on (...)
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  • Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse.Amitayu Banerjee - 2022 - Archive for Mathematical Logic 62 (3):369-399.
    We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of $$\textsf {ZFC}$$ ZFC, then $$\textsf {DC}_{<\kappa }$$ DC < κ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P, G, F ⟩, if $${\mathbb {P}}$$ P is $$\kappa $$ (...)
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  • Provably games.J. P. Aguilera & D. W. Blue - forthcoming - Journal of Symbolic Logic:1-22.
    We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals thatif the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter are determined;all games of length $\omega_1$ with payoff constructible relative to the play are determined; andif the Continuum Hypothesis holds, then there is a model of (...)
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  • More on HOD-supercompactness.Arthur W. Apter, Shoshana Friedman & Gunter Fuchs - 2021 - Annals of Pure and Applied Logic 172 (3):102901.
    We explore Woodin's Universality Theorem and consider to what extent large cardinal properties are transferred into HOD (and other inner models). We also separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. For example, we produce a model where a proper class of supercompact cardinals are not HOD-supercompact but are supercompact in HOD. Additionally we introduce a way to measure the degree of HOD-supercompactness of a supercompact cardinal, and we develop methods to control these degrees simultaneously for a (...)
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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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  • N-Berkeley cardinals and weak extender models.Raffaella Cutolo - 2020 - Journal of Symbolic Logic 85 (2):809-816.
    For a given inner model N of ZFC, one can consider the relativized version of Berkeley cardinals in the context of ZFC, and ask if there can exist an “N-Berkeley cardinal.” In this article we provide a positive answer to this question. Indeed, under the assumption of a supercompact cardinal $\delta $, we show that there exists a ZFC inner model N such that there is a cardinal which is N-Berkeley, even in a strong sense. Further, the involved model N (...)
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  • Dependent choice, properness, and generic absoluteness.David Asperó & Asaf Karagila - forthcoming - Review of Symbolic Logic:1-25.
    We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of (...)
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  • Large cardinals beyond choice.Joan Bagaria, Peter Koellner & W. Hugh Woodin - 2019 - Bulletin of Symbolic Logic 25 (3):283-318.
    The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the (...)
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  • Tame failures of the unique branch hypothesis and models of ADℝ + Θ is regular.Grigor Sargsyan & Nam Trang - 2016 - Journal of Mathematical Logic 16 (2):1650007.
    In this paper, we show that the failure of the unique branch hypothesis for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66 903–923; Core models with more Woodin cardinals, J. Symbolic Logic (...)
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  • Berkeley cardinals and the structure of L.Raffaella Cutolo - 2018 - Journal of Symbolic Logic 83 (4):1457-1476.
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  • Some applications of supercompact extender based forcings to hod.Moti Gitik & Carmi Merimovich - 2018 - Journal of Symbolic Logic 83 (2):461-476.
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  • Generic at.Vincenzo Dimonte - 2018 - Mathematical Logic Quarterly 64 (1-2):118-132.
    In this paper we introduce a generic large cardinal akin to, together with the consequences of being such a generic large cardinal. In this case is Jónsson, and in a choiceless inner model many properties hold that are in contrast with pcf theory in.
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  • Universally baire sets and generic absoluteness.Trevor M. Wilson - 2017 - Journal of Symbolic Logic 82 (4):1229-1251.
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  • In search of ultimate- L the 19th midrasha mathematicae lectures.W. Hugh Woodin - 2017 - Bulletin of Symbolic Logic 23 (1):1-109.
    We give a fairly complete account which first shows that the solution to the inner model problem for one supercompact cardinal will yield an ultimate version ofLand then shows that the various current approaches to inner model theory must be fundamentally altered to provide that solution.
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  • Homogeneous changes in cofinalities with applications to HOD.Omer Ben-Neria & Spencer Unger - 2017 - Journal of Mathematical Logic 17 (2):1750007.
    We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [Formula: see text] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [Formula: see text] in [Formula: see text] is [Formula: see text]-supercompact in HOD.
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  • Two Upper Bounds on Consistency Strength of $negsquare{aleph_{omega}}$ and Stationary Set Reflection at Two Successive $aleph{n}$.Martin Zeman - 2017 - Notre Dame Journal of Formal Logic 58 (3):409-432.
    We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a κ+-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ℵω and make the principle □ℵω,<ω fail in the generic extension. We also (...)
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  • The Weak Ultrafilter Axiom.W. Hugh Woodin - 2016 - Archive for Mathematical Logic 55 (1-2):319-351.
    The main theorem is that the Ultrafilter Axiom of Woodin :115–37, 2011) must fail at all cardinals where the Axiom I0 holds, in all non-strategic extender models subject only to fairly general requirements on the non-strategic extender model.
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  • Equiconsistencies at subcompact cardinals.Itay Neeman & John Steel - 2016 - Archive for Mathematical Logic 55 (1-2):207-238.
    We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □ and □δ fail, then δ is subcompact in a class inner model. If in addition □ fails, we prove that δ is Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary (...)
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  • Laver and set theory.Akihiro Kanamori - 2016 - Archive for Mathematical Logic 55 (1-2):133-164.
    In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.
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  • Collapsing the cardinals of HOD.James Cummings, Sy David Friedman & Mohammad Golshani - 2015 - Journal of Mathematical Logic 15 (2):1550007.
    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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  • The envelope of a pointclass under a local determinacy hypothesis.Trevor M. Wilson - 2015 - Annals of Pure and Applied Logic 166 (10):991-1018.
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  • Consecutive Singular Cardinals and the Continuum Function.Arthur W. Apter & Brent Cody - 2013 - Notre Dame Journal of Formal Logic 54 (2):125-136.
    We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V[G]$ that has a symmetric inner model $N$ in which $\mathrm {ZF}+\lnot\mathrm {AC}$ holds, $\kappa$ and $\kappa^{+}$ are both singular, and the continuum function at $\kappa$ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of $\mathrm (...)
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  • Suitable extender models II: Beyond ω-huge.W. Hugh Woodin - 2011 - Journal of Mathematical Logic 11 (2):115-436.
    We investigate large cardinal axioms beyond the level of ω-huge in context of the universality of the suitable extender models of [Suitable Extender Models I, J. Math. Log.10 101–339]. We show that there is an analog of ADℝ at the level of ω-huge, more precisely the construction of the minimum model of ADℝ generalizes to the level of Vλ+1. This allows us to formulate the indicated generalization of ADℝ and then to prove that if the axiom holds in V at (...)
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