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Almost free groups and Ehrenfeucht-Fraïssé games for successors of singular cardinals [article]

Saharon Shelah, Pauli Väisänen
2002 arXiv   pre-print
We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fraisse games between a fixed group of cardinality lambda and a free Abelian group.  ...  We prove for a large set of successor cardinals lambda=mu^+ existence of nonfree (mu*omega_1)-game-free groups of cardinality lambda. We concentrate on successors of singular cardinals.  ...  For an ordinal ǫ, a group G is called ǫ-game-free, if the "isomorphism player" has a winning strategy in the Ehrenfeucht-Fraïssé game of length ǫ between G and a free Abelian group (countable many elements  ... 
arXiv:math/0212063v1 fatcat:ozvuznpggbejtnzadwajefjfii

Almost free groups and Ehrenfeucht–Fraı̈ssé games for successors of singular cardinals

Saharon Shelah, Pauli Väisänen
2002 Annals of Pure and Applied Logic  
We strengthen nonstructure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fra ssà e games between a ÿxed group of cardinality and a free Abelian group.  ...  We prove for a large set of successor cardinals = + the existence of nonfree ( · !1)-game-free groups of cardinality . We concentrate on successors of singular cardinals. (S.  ...  On cub-game and game-free groups In this section, is a singular cardinal, Ä denotes the coÿnality of , and is the successor cardinal of .  ... 
doi:10.1016/s0168-0072(02)00037-4 fatcat:upvdjksw7rg7jhuh5btpf5h3by

Page 5727 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
Stoyan Nedev (BG-AOS; Sofia) 2003h:03073 03E05 03C75 20K20 Shelah, Saharon (IL-HEBR-IM; Jerusalem) ; Vaisinen, Pauli (FIN-HELS; Helsinki) Almost free groups and Ehrenfeucht-Fraissé games for successors  ...  A /- free group of cardinality / is called almost free. The first author proved that an almost free group of singular cardinality is free.  ... 

Weak Ehrenfeucht-Fraïssé games

Tapani Hyttinen, Vadim Kulikov
2011 Transactions of the American Mathematical Society  
We investigate determinacy of the weak EF-game for different α (the length of the game) and its relation to the ordinary EF-game.  ...  Thus the difference between this and the ordinary Ehrenfeucht-Fraïssé game is that the isomorphism can be arbitrary, whereas in the ordinary EF-game it is determined by the moves of the players.  ...  An almost free group is such a group that all its countable subgroups (or more generally, all subgroups of size < κ for κ an uncountable cardinal) are free, and similarly for free and almost free abelian  ... 
doi:10.1090/s0002-9947-2011-05222-0 fatcat:xshduwvfknhahicyfm56tt5mpy

The Ehrenfeucht-Fraisse-game of length omega_1 [article]

Alan H. Mekler, Saharon Shelah, Jouko Väänänen
1993 arXiv   pre-print
This game is like the ordinary Ehrenfeucht-Fraisse-game of L_omega omega except that there are omega_1 moves. It is clear that G_omega_1(A,B) is determined if A and B are of cardinality <= aleph_1.  ...  Let (A) and (B) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraisse-game of length omega_1 of A and B which we denote by G_omega_1(A,B).  ...  Therefore, if G ω 1 (A, B) is determined for all almost free groups A and B of cardinality ℵ 2 , then ω 2 is Mahlo in L.  ... 
arXiv:math/9305204v1 fatcat:pcjkyiee6jaejapovvti2zgefu

The Ehrenfeucht-Fraisse-Game of Length ω 1

Alan Mekler, Saharon Shelah, Jouko Vaananen
1993 Transactions of the American Mathematical Society  
This game is like the ordinary Ehrenfeucht-Fraïssé-game of L ωω except that there are ω 1 moves. It is clear that G ω 1 (A, B) is determined if A and B are of cardinality ≤ ℵ 1 .  ...  Let A and B be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraïssé-game of length ω 1 of A and B which we denote by G ω 1 (A, B) .  ...  Therefore, if G ω 1 (A, B) is determined for all almost free groups A and B of cardinality ℵ 2 , then ω 2 is Mahlo in L.  ... 
doi:10.2307/2154287 fatcat:o5fevwx25zggvbovajp5pzcxmi

Constructing strongly equivalent nonisomorphic models for unstable theories

Tapani Hyttinen, Heikki Tuuri
1991 Annals of Pure and Applied Logic  
If T is an unstable theory of cardinality o (d > w, in the superstable with DOP case) is regular and A<* = h, then we construct for T strongly equivalent nonisomorphic models of cardinality 1.  ...  Hyttinen, T. and H. Tuuri, Constructing strongly equivalent nonisomorphic models for unstable theories, Annals of Pure and Applied Logic 52 (1991) 203-248.  ...  The authors would also like to thank Juha Oikkonen, Taneli Huuskonen and Lauri Hella for their insights and inspiring discussions, Bradd Hart for his valuable help and Daniel Lascar for teaching stability  ... 
doi:10.1016/0168-0072(91)90031-g fatcat:badcvfs4jbgk7ishcpvxbng6cy

Laver and set theory

Akihiro Kanamori
2016 Archive for Mathematical Logic  
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.  ...  Doctoral students of Richard Laver Stephen Grantham, An analysis of Galvin In addition to having these doctoral students at Boulder, Laver was on the thesis committees of, among many:  ...  Then in 1947 Roland Fraïssé, now best known for the Ehrenfeucht-Fraïssé games and Fraïssé limits, pointed to basic issues for countable order types in four conjectures.  ... 
doi:10.1007/s00153-015-0462-7 fatcat:5ziwvh27arbprhsdlmku4f2npe

Page 107 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
(English summary) 2003e:03091 Shelah, Saharon (with Vaisénen, Pauli) Almost free groups and Ehrenfeucht-Fraissé games for successors of singular cardinals.  ...  .) * Almost free modules. 2003e:20002 Goldstern, Martin R. (with Shelah, Saharon) Clones on regular cardinals.  ... 

Page 2275 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
(From the text) 2004g:03072 03504 AUTHOR INDEX Shelah — (with Hyttinen, Tapani; Vaananen, Jouko) More on the Ehrenfeucht-Fraissé game of length w;. (English summary) Fund.  ...  (Howard Smith) 2004j:20074 20F19 (20E25) — Successor of singulars: combinatorics and not collapsing cardinals < « in (< «)-support iterations. (English summary) /srael J. Math. 134 (2003), 127-155.  ... 

Algebraic logic and topoi; a philosophical holistic approach

Tarek Sayed Ahmed
2020 International Journal of Algebra  
For example topoi and category theory are approached as means to unify apparently scattered results in the literature.  ...  the number of subvarieties of RCA ω which is reminiscent of Shelah's stability theory and the interaction of algebraic logic and descriptive set theory as means to approach Vaught's conjecture in model  ...  his winning strategy from the private Ehrenfeucht-Fraïssé forth game, to the graph game on At f = At(A n+1,n ) [26, p 841] forcing a win using 2n nodes.  ... 
doi:10.12988/ija.2020.91249 fatcat:razdk5433fba5hrg6p6wb65a4y

Page 1899 of Mathematical Reviews Vol. , Issue Index [page]

Mathematical Reviews  
(Judith Roitman) 2003c:03091 03E40 (03G05, 06E05) — (with Vaisanen, Pauli) Almost free groups and Ehrenfeucht-Fraissé games for successors of singular cardinals. (English summary) Ann. Pure Appl.  ...  A QFT procedure for generating design frequencies and bounds of guaranteed accuracy. (English summary) Reliab.  ... 

Can we classify complete metric spaces up to isometry? [article]

Luca Motto Ros
2017 arXiv   pre-print
The results concerning non-separable spaces (and, to some extent, the setup and techniques used to handle them) are instead new, and suggest new lines of investigation in this area of research.  ...  We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others.  ...  κ-branches, and winning strategies in the corresponding dynamic Ehrenfeucht-Fraïssé games (this yields a neat analog of the notion of Scott watershed from the Scott analysis of countable models, see  ... 
arXiv:1610.01750v2 fatcat:vcyvmnnygfdw5ctzpss4azplpe

The structure of graphs and new logics for the characterization of Polynomial Time [article]

Bastian Laubner, Humboldt-Universität Zu Berlin, Humboldt-Universität Zu Berlin
2017
Our exposition provides evidence that rank logics are a natural object to study and establishes the most expressive of our rank logics as a viable candidate for capturing PTIME, suggesting that rank logics  ...  First, we adapt a representation-invariant, singly exponential-time graph canonization algorithm of Corneil and Goldberg (1984) and conclude that on structures whose relations are of arity at most 2, the  ...  Ehrenfeucht-Fraïssé style pebble games, for example, are typically played by two players: spoiler and duplicator.  ... 
doi:10.18452/16335 fatcat:f6rgrxx2hfghdfse7o75hpai7a