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A model and its subset: the uncountable case

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*. ...

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

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*. ...

##
###
Distinct Volume Subsets via Indiscernibles
[article]

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. ...

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

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 ...

##
###
Forcing closed unbounded subsets of ω2

2001
*
Annals of Pure and Applied Logic
*

This paper addresses

doi:10.1016/s0168-0072(00)00062-2
fatcat:k7rsvoyoefgjngeprkcov7dw7a
*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). ...##
###
On the free subset property at singular cardinals

1989
*
Archive for Mathematical Logic
*

We give

doi:10.1007/bf01624082
fatcat:wf26pbyj3bfzvmzwv2dy5b3ywm
*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(~:). ...##
###
On coverings of Banach spaces and their subsets by hyperplanes
[article]

2021
*
arXiv
*
pre-print

Given

arXiv:2103.05097v2
fatcat:rocgfzg7v5fqljfw6e5s4kt35y
*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*a*∈*A*′ .*It*follows that*the*closure of*A*′ is disjoint from diagonal as g|∆(K) = 0 which completes*the*proof of (3). ...##
###
Stein surfaces as open subsets of C^2
[article]

2006
*
arXiv
*
pre-print

An open

arXiv:math/0501509v3
fatcat:purmljvh3ng7zlfponfher6wly
*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 ...##
###
Partitioning subsets of stable models

2001
*
Journal of Symbolic Logic (JSL)
*

First we prove

doi:10.2307/2694983
fatcat:gf5riu3jenbwzn72ucqcljf6ay
*a*partition result for*subsets*of stable*models*: for any*A**and*B, we can partition*A*into ∣B∣<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*. ...##
###
On strongly discrete subsets of $\omega\sp \ast$

1993
*
Proceedings of the American Mathematical Society
*

We prove that

doi:10.1090/s0002-9939-1993-1181172-7
fatcat:stwivgapmfe7rcbgrynz42hrvi
*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? ...##
###
Projective subsets of separable metric spaces

1990
*
Annals of Pure and Applied Logic
*

there exists

doi:10.1016/0168-0072(90)90054-6
fatcat:o3taesduovcqbfrqzfespgz5ze
*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. ...##
###
On translations of subsets of the real line

2001
*
Proceedings of the American Mathematical Society
*

.,

doi:10.1090/s0002-9939-01-06224-4
fatcat:zkejd6ug5bhhtkankqqw3lagqq
*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*. ...##
###
Perfect subsets of generalized Baire spaces and long games
[article]

2017
*
arXiv
*
pre-print

In

arXiv:1703.10148v2
fatcat:5xfsmyeytbfwjm7cmbvnmclz3e
*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 ). ...##
###
Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC
[article]

2021
*
arXiv
*
pre-print

If in

arXiv:2009.05368v2
fatcat:3g4ppidn7na2dk5cgthscnmqui
*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 . ...##
###
A non-compact deduction rule for the logic of provability and its algebraic models
[article]

2020
*
arXiv
*
pre-print

In this paper, we introduce

arXiv:2002.04782v5
fatcat:jqmys7n3nvhnzj5vi2w5zqypmm
*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. ...
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