Large Cardinals from Determinacy

Koellner, Peter; Woodin, W. Hugh

doi:10.1007/978-1-4020-5764-9_24

Cited by 96 publications

(93 citation statements)

References 7 publications

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“…Conversely, it is a basic result of Woodin that assuming AD + , every Suslin and co-Suslin set A is captured by some (M, Σ) such that Σ has the Dodd-Jensen property. In fact Theorem 10.2 (Woodin, see [5] and [23].) Assume AD + , let Γ be a good scaled pointclass not closed under ∀ R , and let A ∈ Γ.…”

Section: The Background Universe N * Xmentioning

confidence: 99%

Derived Models Associated to Mice

Steel

2008

Computational Prospects of Infinity

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We shall present some results on the the properties of derived models of mice, and on the existence of mice with large derived models. The basic plan of the paper is:Sections 1-5. Introduction. Preliminary definitions and background material. The iteration independence of the derived model associated to a mouse. The mouse set conjecture in derived models associated to mice.Sections 6-8 The Solovay sequence in derived models associated to mice. A method of "translating away" extenders overlapping Woodin cardinals in mice.Sections 9-14. Some partial results on capturing Σ 2 1 truths in AD + models by mice. (That is, partial results on the mouse set conjecture.)Section 15 A theorem of Woodin on the consistency strength of AD + + θ 0 < θ. Section 16The global mouse set conjecture implies its local version.Section 17 Capturing sets of reals in R-mice.

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Section: The Background Universe N * Xmentioning

confidence: 99%

Derived Models Associated to Mice

Steel

2008

Computational Prospects of Infinity

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“…These axioms are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent (Kanamori, 2003;Koellner & Woodin, 2010;Shelah & Woodin, 1990). Going further, a cardinal κ is superstrong if there is a transitive class M and a non-trivial elementary embedding j : V → M such that crit(j) = κ and V j(κ) ⊆ M. If κ is super strong then k is a Woodin cardinal and there are arbitrarily large Woodin cardinals below κ (Riis, 1994).…”

Section: Concepts In the Theory Of Setsmentioning

confidence: 99%

“…8, No. 6; prospect for the construction of an ultimate enlargement of L (Koellner & Woodin, 2010;Shelah & Woodin, 1990;Gödel, 1939).…”

Section: Concepts In the Theory Of Setsmentioning

confidence: 99%

Some Proposition that Links Ferromagnetic Models with Cantorian Set Theory

Gebril¹

2016

APR

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In this paper, we established a link between ferromagnetic models and set theory. Three dimension spins "S X , S y , S z " are considered in Heisenberg model but it is restricted to z_ axis " S z " in Ising mode. By using Brouwer theory, fractal motion of magnetic domains is predicted theoretically in the materials that their magnetism are explained by Ising model. In addition, we achieved that the experimental data of the fractal motion of magnetic domains in the thin films are agreement with the theoretical assumptions that are proposed.

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“…The axiom AD + is a strengthening of the Axiom of Determinacy due to Woodin (for this and the two following theorems, see [7]). We will be using the following well-known facts, though some of them are unpublished.…”

Section: Proposition 39 In the Situation Of Theorem 32 Assuming ♦mentioning

confidence: 99%

Absoluteness for Universally Baire Sets and the Uncountable Ii

Farah¹,

Ketchersid²,

Larson³

et al. 2008

Computational Prospects of Infinity

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Abstract. Using ♦ and large cardinals we extend results of Magidor-Malitz and Farah-Larson to obtain models correct for the existence of uncountable homogeneous sets for finite-dimensional partitions and universally Baire sets. Furthermore, we show that the constructions in this paper and its predecessor can be modified to produce a family of 2 ω 1 -many such models so that no two have a stationary, costationary subset of ω 1 in common. Finally, we extend a result of Steel to show that trees on reals of height ω 1 which are coded by universally Baire sets have either an uncountable path or an absolute impediment preventing one.In [3] it was shown (using large cardinals) that if a model of a theory T satisfying a certain second-order property P can be forced to exist, then a model of T satisfying P exists already. The properties P considered in [3] included the following.(1) Containing any specified set of ℵ 1 -many reals.(2) Correctness about NS ω 1 . (3) Correctness about any given universally Baire set of reals (with a predicate for this set added to the language). In this paper we add the following properties, all proved under the assumption of Jensen's ♦ principle. (1) and (7), we can obtain all of these properties simultaneously using the method (a) (with "some" being "all" for (5) and (6)). Aside from (1) and (4) we can prove all of these properties simultaneously using the method (b). Property (4)

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Large Cardinals from Determinacy

Cited by 96 publications

References 7 publications

Derived Models Associated to Mice

Derived Models Associated to Mice

Some Proposition that Links Ferromagnetic Models with Cantorian Set Theory

Absoluteness for Universally Baire Sets and the Uncountable Ii

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