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Transfer Principles in Henselian Valued Fields

Pierre Touchard
2021 Bulletin of Symbolic Logic  
Second, we show a 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.  ... 
doi:10.1017/bsl.2021.31 fatcat:cw6v3vw4fbbfjn5w5shhs5xteu

Categoricity transfer for short AECs with amalgamation over sets [article]

Samson Leung
2022 arXiv   pre-print
Suppose K is categorical 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.  ... 
arXiv:2203.08956v1 fatcat:racxdwyzsfe5zcro7h2wov2pry

Transfering saturation, the finite cover property, and stability

John T. Baldwin, Rami Grossberg, Saharon Shelah
1999 Journal of Symbolic Logic (JSL)  
Saturation is (μ, κ)-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 |.  ... 
doi:10.2307/2586492 fatcat:2gjuiywsrfc4zjifxhrqit7rjm

Transferring saturation, the finite cover property, and stability [article]

J. Baldwin, R. Grossberg, Saharon Shelah
1998 arXiv   pre-print
Saturation is (mu,kappa)-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|.  ... 
arXiv:math/9511205v3 fatcat:f7a5to3xzvevznb3u4bqk57msy

Models with dimension

W.F. Gross
1976 Bulletin of the Australian Mathematical Society  
We then investigate whether our 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.  ... 
doi:10.1017/s0004972700024953 fatcat:6f5a6m6kjneatprsmf6ls6lvca

Contributions to the Theory of Large Cardinals through the Method of Forcing

Alejandro Poveda
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.  ... 
doi:10.1017/bsl.2021.22 fatcat:yynwd7jnyfc2vmkgazoqknnvua

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

Rami Grossberg, Monica VanDieren
2005 arXiv   pre-print
MAIN COROLLARY: (ZFC) If K is categorical 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.  ... 
arXiv:math/0509387v2 fatcat:ky5nwoezffdcjj3prdboowmfnu

Some downwards transfer properties for N2

Matthew Foreman, Richard Laver
1988 Advances in Mathematics  
INTRODUCTION Assuming the consistency of ZFC + "there is a huge 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 ?  ... 
doi:10.1016/0001-8708(88)90041-2 fatcat:72mxmf5pdncunhhnm6p4gjmede

Shelah's categoricity conjecture from a successor for tame abstract elementary classes

Rami Grossberg, Monica Vandieren
2006 Journal of Symbolic Logic (JSL)  
has arbitrarily large 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.  ... 
doi:10.2178/jsl/1146620158 fatcat:7rm3fisf4zbkbbipa3o4gwyijy

Transferring Symmetry [article]

Monica VanDieren
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 + .  ... 
arXiv:1507.01991v1 fatcat:rvjlglfwkvc6lfbjibhwtr4fyu

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  ... 

Symmetry and the union of saturated models in superstable abstract elementary classes

M.M. VanDieren
2016 Annals of Pure and Applied Logic  
Theoerem 1: Let K be an abstract elementary class with no maximal 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.  ... 
doi:10.1016/j.apal.2015.12.007 fatcat:jvow3qdrb5cupbghgk4ym3z7xi

Symmetry in abstract elementary classes with amalgamation

Monica M. VanDieren, Sebastien Vasey
2017 Archive for Mathematical Logic  
These results are then used to prove several structural 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).  ... 
doi:10.1007/s00153-017-0533-z fatcat:jxakzt5aozfgrjmhrmla3px26m

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.  ... 

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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