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A model and its subset: the uncountable case

Ludomir Newelski
1995 Annals of Pure and Applied Logic  
The classical theorem of Karp [6] says that two models M, N are L"-equivalent iff there is a back-and-forth relation -between M and N.  ...  Assume Q is a definable subset of a model of T. We define a notion of Q-isolated type, generalizing an earlier definition for countable Q. This notion is absolute.  ...  Introduction This paper is a continuation of the study of the relationship between a model and its definable subset, carried in [9, lo] . In [9, lo] we considered mainly countable models.  ... 
doi:10.1016/0168-0072(95)91363-f fatcat:fo75nzoqsfdq3hkbbpwnjl5u3u

Phase transitions for a model with uncountable spin space on the Cayley tree: the general case [article]

Golibjon Botirov, Benedikt Jahnel
2018 arXiv   pre-print
The potential is of nearest-neighbor type and the local state space is compact but uncountable.  ...  In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12,EsRo10,BoEsRo13,JaKuBo14,Bo17].  ...  It is the purpose of this note to complete the analysis of this model.  ... 
arXiv:1803.02867v1 fatcat:xik3tle3rreb3jxctjpnxjb2nm

Distinct Volume Subsets via Indiscernibles [article]

William Gasarch, Douglas Ulrich
2018 arXiv   pre-print
The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, Δ is a finite set of formulas of T, M T, and X is an infinite subset of M.  ...  In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery.  ...  We also want to thank our co-authors on the original Distinct Volume Sets paper, (David Conlon, Jacob Fox, David Harris, and Sam Zbarsky) since that paper was also one of our inspirations.  ... 
arXiv:1807.06654v1 fatcat:5hjzfysbjfanlk5pzb5cpke7xa

On the existence of large subsets of [lambda]^<kappa which contain no unbounded non-stationary subsets [article]

Saharon Shelah
1999 arXiv   pre-print
The first section deals with the existence of stationary subsets of [lambda]^<kappa with no unbounded subsets which are not stationary, where, of course, kappa is regular uncountable less or equal than  ...  Theorem 1.2 was proved some time ago by Baumgartner and is presented here for the sake of completeness.  ...  a α : α < 2 λ belong, each N ζ is of the cardinality of the continuum, θ a regular uncountable ≤ 2 ℵ 0 and N ǫ : ǫ ≤ ζ ∈ N ζ+1 for ζ < θ (note that N 1+ζ ∩ κ ∈ κ follows in this case and for limit ζ in  ... 
arXiv:math/9908159v1 fatcat:deadat2smjaqncwtvuthhepetq

Forcing closed unbounded subsets of ω2

M.C. Stanley
2001 Annals of Pure and Applied Logic  
This paper addresses a special case of this problem when κ is singular, namely, the case of "bounded pattern width" subsets of ℵ ω+1 .  ...  For (1) ⇒ (2), suppose that C ⊆ X is a club subset of κ + lying in an outer model of V as in (1). In this model κ remains regular and uncountable.  ...  It follows that X has a club subset in a further GCH, κ, and κ + preserving set generic extension. Thus (3) follows from (2).  ... 
doi:10.1016/s0168-0072(00)00062-2 fatcat:k7rsvoyoefgjngeprkcov7dw7a

On the free subset property at singular cardinals

Peter Koepke
1989 Archive for Mathematical Logic  
We give a proof of Theorem 1. Let x be the smallest cardinal such that the free subset property Fr~,(x, coO holds. Assume ~ is singular. Then there is an inner model with co~ measurable cardinals.  ...  Let Z be an uncountable free subset for the structure K~+. We consider transitive collapses/~r of the substructures of K~ + generated by uncountable subsets Y of Z.  ...  We conclude the proof of Theorem 1 according to two cases: Case 1. There exists an uncountable X__c Z such that fl(X)= col and {2x[i < col} is cofinal in g = az(~:).  ... 
doi:10.1007/bf01624082 fatcat:wf26pbyj3bfzvmzwv2dy5b3ywm

On coverings of Banach spaces and their subsets by hyperplanes [article]

Damian Głodkowski, Piotr Koszmider
2021 arXiv   pre-print
Given a Banach space we consider the σ-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering  ...  The remaining questions reduce to deciding if the following can be proved in ZFC for every nonseparable Banach space X: (1) X can be covered by ω_1-many of its hyperplanes; (2) All subsets of X of cardinalities  ...  Hence there exist ε > 0 and an uncountable subset A ′ ⊆ A such that g(a) > ε for aA ′ . It follows that the closure of A ′ is disjoint from diagonal as g|∆(K) = 0 which completes the proof of (3).  ... 
arXiv:2103.05097v2 fatcat:rocgfzg7v5fqljfw6e5s4kt35y

Stein surfaces as open subsets of C^2 [article]

Robert E. Gompf
2006 arXiv   pre-print
An open subset U of a complex surface can be topologically perturbed to yield an open subset whose inherited complex structure is Stein, if and only if U is homeomorphic to the interior of a handlebody  ...  The argument of [DF] is too specialized to apply to all Stein surfaces, but in some cases (e.g., any open subset of C 2 ) we can distinguish uncountably many diffeomorphism types after we connect by  ...  There is a family of open subsets of C 2 (with compact closure) that are Stein and homeomorphic to R 4 , but realize uncountably many diffeomorphism types (with the cardinality of the continuum in ZFC  ... 
arXiv:math/0501509v3 fatcat:purmljvh3ng7zlfponfher6wly

Partitioning subsets of stable models

Timothy Bays
2001 Journal of Symbolic Logic (JSL)  
First we prove a partition result for subsets of stable models: for any A and B, we can partition A into ∣B∣&lt;κ(T) pieces.  ...  ∣B′∣ is as small as possible, and . We prove some positive results in this direction, and we discuss the optimality of these results under ZFC + GCH.  ...  The main theorem in the section is the following: Theorem: Let A and B be arbitrary subsets of a stable model.  ... 
doi:10.2307/2694983 fatcat:gf5riu3jenbwzn72ucqcljf6ay

On strongly discrete subsets of $\omega\sp \ast$

Mariusz Rabus
1993 Proceedings of the American Mathematical Society  
We prove that it is consistent with Martin's Axiom and -CH that there is a strongly discrete subspace A C co* of cardinality N] such that the closure of A is not homeomorphic with piox .  ...  We also prove that MA and -iCH imply that there is no convergent strongly discrete subset of co* .  ...  As pointed out in the introduction it is enough to construct a model of MA with subsets & = {pa : a £ cox} and 2? = {qa : a £ cox} of co* such that &~ 1)2?  ... 
doi:10.1090/s0002-9939-1993-1181172-7 fatcat:stwivgapmfe7rcbgrynz42hrvi

Projective subsets of separable metric spaces

Arnold W. Miller
1990 Annals of Pure and Applied Logic  
there exists a complete separable metric space Y and a Bore1 B E X x Y such that A = proj,(B) = {x E X: 3y E Y (x, y) E B}; (2) (relatively analytic) if 8 zk the completion of X, then there exists a E  ...  2, a 2:(R), set such that A = a fl X; * Partially supported by NSF grant 8801139. 0168~0072/90/$03.50 0 1990 -Elsevier Science Publishers B.V.  ...  Properties of products A separable metrix space X is Luzin iff it is uncountable and every meager subset of X is countable.  ... 
doi:10.1016/0168-0072(90)90054-6 fatcat:o3taesduovcqbfrqzfespgz5ze

On translations of subsets of the real line

Jacek Cichoń, Andrzej Jasiński, Anastasis Kamburelis, Przemysław Szczepaniak
2001 Proceedings of the American Mathematical Society  
., it is closed under taking unions of length less than add(J). If A ∈ J, then obviously A and A c are J-almost invariant. We consider such sets as trivial J-almost invariant sets.  ...  Translations of subsets of the real line In this section we discuss some translation properties of subsets of the real line R. Hence our basic group is the real line and the basic ideal is {∅}.  ...  Thus V [c] models the absolute sentence "(∃x)((A * α − x) ∩ B * β is uncountable)" and we conclude that (in V ) there exists x such that (A − x) ∩ B is uncountable.  ... 
doi:10.1090/s0002-9939-01-06224-4 fatcat:zkejd6ug5bhhtkankqqw3lagqq

Perfect subsets of generalized Baire spaces and long games [article]

Philipp Schlicht
2017 arXiv   pre-print
In the second main theorem, we introduce a Banach-Mazur type game of length λ and show that the determinacy of this game, for all subsets of ^λλ that are definable from elements of ^λOrd as winning conditions  ...  We extend Solovay's theorem about definable subsets of the Baire space to the generalized Baire space ^λλ, where λ is an uncountable cardinal with λ^<λ=λ.  ...  For instance, it is consistent relative to the existence of an inaccessible cardinal that this is the case in the λ-Chang model C λ = L(Ord λ ).  ... 
arXiv:1703.10148v2 fatcat:5xfsmyeytbfwjm7cmbvnmclz3e

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC [article]

Amitayu Banerjee
2021 arXiv   pre-print
If in a partially ordered set all antichains are finite and all chains have size ℵ_α, then the set has size ℵ_α if ℵ_α is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5.  ...  In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1.  ...  Clearly, ≤ is a partial order on A. Also, A is uncountable. The only antichains of (A, ≤) are the finite sets A n and subsets of A n where n ∈ ℵ 1 .  ... 
arXiv:2009.05368v2 fatcat:3g4ppidn7na2dk5cgthscnmqui

A non-compact deduction rule for the logic of provability and its algebraic models [article]

Yoshihito Tanaka
2020 arXiv   pre-print
In this paper, we introduce a proof system with a non-compact deduction rule, that is, a deduction rule with countably many premises, to axiomatize the logic GL of provability, and show its Kripke completeness  ...  As GL is not canonical, a standard proof of Kripke completeness for GL is given by a Kripke model which is obtained by changing the binary relation of the canonical model, while our proof is given by a  ...  Let M = (W, R, v) be a Kripke model and φ a formula.  ... 
arXiv:2002.04782v5 fatcat:jqmys7n3nvhnzj5vi2w5zqypmm
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