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Transfer Principles in Henselian Valued Fields

2021
*
Bulletin of Symbolic Logic
*

Second, we show a

doi:10.1017/bsl.2021.31
fatcat:cw6v3vw4fbbfjn5w5shhs5xteu
*transfer*principle for the*property*that all types realized*in*a given elementary extension are definable. ... The burden is a*cardinal*related to the*model*theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. ... Second, we show a*transfer*principle for the*property*that all types realized*in*a given elementary extension are definable. ...##
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Categoricity transfer for short AECs with amalgamation over sets
[article]

2022
*
arXiv
*
pre-print

Suppose K is categorical

arXiv:2203.08956v1
fatcat:racxdwyzsfe5zcro7h2wov2pry
*in*some μ>LS( K), then it is categorical*in*all μ'≥μ. ... As a corollary, we obtain an alternative proof of the upward categoricity*transfer*for first-order theories by Morley and Shelah. ...*transfers*up from an uncountable*cardinal*. ...##
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Transfering saturation, the finite cover property, and stability

1999
*
Journal of Symbolic Logic (JSL)
*

Saturation is (μ, κ)-

doi:10.2307/2586492
fatcat:2gjuiywsrfc4zjifxhrqit7rjm
*transferable**in*T if and only if there is an expansion T 1 of T with |T 1| = |T| such that if M is a μ-saturated*model*of T 1 and |M| ≥ κ then the reduct M|L(T) is κ-saturated. ... ., or without f.c.p. as, respectively those where saturation is (ℵ0, λ)-*transferable*or (κ(T), λ)-*transferable*for all λ. ... (over A)*extending*I of*cardinality*|M |. ...##
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Transferring saturation, the finite cover property, and stability
[article]

1998
*
arXiv
*
pre-print

Saturation is (mu,kappa)-

arXiv:math/9511205v3
fatcat:f7a5to3xzvevznb3u4bqk57msy
*transferable**in*T if and only if there is an expansion T_1 of T with |T_1| = |T| such that if M is a mu-saturated*model*of T_1 and |M| \geq kappa then the reduct M|L(T) is kappa-saturated ... We characterize theories which are superstable without the finite cover*property*(f.c.p.), or without f.c.p. as, respectively those where saturation is (aleph_0,lambda)-*transferable*or (kappa(T),lambda ... (over A)*extending*I of*cardinality*|M|. ...##
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Models with dimension

1976
*
Bulletin of the Australian Mathematical Society
*

We then investigate whether our

doi:10.1017/s0004972700024953
fatcat:6f5a6m6kjneatprsmf6ls6lvca
*models*have the*properties*mentioned*in*the first paragraph*in*the situations of not assuming AC and of assuming the axiom of choice for sets of finite sets (ACF). ... It is well known that, if AC is assumed, then any vector space has a basis, any independent subset can be*extended*to a basis, any two bases of a vector space have the same*cardinality*and a one-one map ... However,*in*the order Mostowski*model*of ZFA + ACF , for any*model*M , all bases are of the same*cardinality*. ...##
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Contributions to the Theory of Large Cardinals through the Method of Forcing

2021
*
Bulletin of Symbolic Logic
*

*In*particular, if Woodin's HOD Conjecture holds, and therefore it is provable

*in*ZFC + "There exists an

*extendible*

*cardinal*" that above the first

*extendible*

*cardinal*every singular

*cardinal*$\lambda $ ... Two of these are the Tree

*Property*and the Reflection of Stationary sets, which are central

*in*Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree

*Property*at both ... Second, we show a

*transfer*principle for the

*property*that all types realized

*in*a given elementary extension are definable. ...

##
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Shelah's Categoricity Conjecture from a successor for Tame Abstract Elementary Classes
[article]

2005
*
arXiv
*
pre-print

MAIN COROLLARY: (ZFC) If K is categorical

arXiv:math/0509387v2
fatcat:ky5nwoezffdcjj3prdboowmfnu
*in*a successor*cardinal*bigger than _(2^μ)^+ then K is categorical*in*all*cardinals*greater than _(2^μ)^+. ... Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding*property*, let μ be the Hanf number of K. Suppose K is tame. ... If K is categorical*in*λ and λ + , then every rooted minimal type over a*model*N of*cardinality*λ + is realized λ ++ times*in*every*model*of*cardinality*λ ++*extending*N . Proof. ...##
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Some downwards transfer properties for N2

1988
*
Advances in Mathematics
*

INTRODUCTION Assuming the consistency of ZFC + "there is a huge

doi:10.1016/0001-8708(88)90041-2
fatcat:72mxmf5pdncunhhnm6p4gjmede
*cardinal*," we construct a*model*of ZFC + GCH*in*which the*cardinal*N, possesses some downwards*transfer**properties*. ... The*model*of this paper has downwards*transfer**properties*on y + + as well as a ? ...##
###
Shelah's categoricity conjecture from a successor for tame abstract elementary classes

2006
*
Journal of Symbolic Logic (JSL)
*

has arbitrarily large

doi:10.2178/jsl/1146620158
fatcat:7rm3fisf4zbkbbipa3o4gwyijy
*models*. ... AbstractWe prove a categoricity*transfer*theorem for tame abstract elementary classes.Suppose thatKis a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding*properties*and ... If K is categorical*in*λ and λ + , then every rooted minimal type over a*model*N of*cardinality*λ + is realized λ ++ times*in*every*model*of*cardinality*λ ++*extending*N . Proof. ...##
###
Transferring Symmetry
[article]

2015
*
arXiv
*
pre-print

*In*this paper, we apply results of Va3 and use towers to

*transfer*symmetry from μ^+ down to μ

*in*superstable abstract elementary classes without using extra set-theoretic assumptions or tameness. ... Suppose K is an abstract elementary class satisfying the amalgamation and joint embedding

*properties*and that K is both μ- and μ^+-superstable. ...

*In*the proof of Theorem 0.1, we will be using towers composed of

*models*of

*cardinality*µ and other towers composed of

*models*of

*cardinality*µ + . ...

##
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Page 3148 of Mathematical Reviews Vol. , Issue 2000e
[page]

2000
*
Mathematical Reviews
*

The sophisticated, “ elegant technique used

*in*the proof of the*transfer*theorem has —.2900e:03144 03E55 roots*in*work of Shelah on*models*with second-order*properties*. ... The existence of . 4 ‘ ed*cardinal*is a strong large*cardinal*—— it aes*Models*, algebras, and pros (Bogota, 1995), 51-56, Lecture Notes known one from which significant relative consistency results*in*...##
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Symmetry and the union of saturated models in superstable abstract elementary classes

2016
*
Annals of Pure and Applied Logic
*

Theoerem 1: Let K be an abstract elementary class with no maximal

doi:10.1016/j.apal.2015.12.007
fatcat:jvow3qdrb5cupbghgk4ym3z7xi
*models*of*cardinality*μ^+ which satisfies the joint embedding and amalgamation*properties*. Suppose μ≥ LS(K). ... We also apply results of VanDieren's Superstability and Symmetry paper and use towers to*transfer*symmetry from μ^+ down to μ*in*abstract elementary classes which are both μ- and μ^+-superstable: Theorem ... Acknowledgement The author is grateful to Sebastien Vasey for email correspondence about [1] during which he asked her about the union of saturated*models*. ...##
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Symmetry in abstract elementary classes with amalgamation

2017
*
Archive for Mathematical Logic
*

These results are then used to prove several structural

doi:10.1007/s00153-017-0533-z
fatcat:jxakzt5aozfgrjmhrmla3px26m
*properties**in*categorical AECs, improving classical results of Shelah who focused on the special case of categoricity*in*a successor*cardinal*. ... The key results are a downward*transfer*of symmetry and a deduction of symmetry from failure of the order*property*. ... This shows that q*extends*p, as desired. We can now prove an extension*property*for towers*in*K * λ,α,µ . Lemma 3.5. Let λ and µ be*cardinals*satisfying λ ≥ µ ≥ LS(K). ...##
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Page 1127 of Mathematical Reviews Vol. 47, Issue 5
[page]

1974
*
Mathematical Reviews
*

= a has a

*model*of*cardinal*&, ,,, whose universe contains X and is such that any order automorphism of X can be*extended*to an automorphism of the*model*. ...*Transfer*theorems involving hyperinaccessible*cardinals*for*models*of elementary theories and theories that allow the omission of types are the main topic of the paper. ...##
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Page 6435 of Mathematical Reviews Vol. , Issue 96k
[page]

1996
*
Mathematical Reviews
*

This paper is devoted to the study of the strength of the

*model*- theoretic*transfer**property*(XN), No) — (A*,A). ...*transfer**property*and resurrection of supercompactness. ...
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