More on the Ehrenfencht-Fraisse game of length omega_1
| dc.creator | Hyttinen, Tapani | |
| dc.creator | Shelah, Saharon | |
| dc.creator | Väänänen, Jouko | |
| dc.date | 2002-12-17 | |
| dc.date.accessioned | 2026-07-07T04:53:51Z | |
| dc.date.available | 2026-07-07T04:53:51Z | |
| dc.description | Let A and B be two first order structures of the same relational vocabulary L. The Ehrenfeucht-Fraisse-game of length gamma of A and B denoted by EFG_gamma(A,B) is defined as follows: There are two players called for all and exists. First for all plays x_0 and then exists plays y_0. After this for all plays x_1, and exists plays y_1, and so on. Eventually a sequence <(x_beta,y_beta): beta<gamma> has been played. The rules of the game say that both players have to play elements of A cup B. Moreover, if for all plays his x_beta in A (B), then exists has to play his y_beta in B (A). Thus the sequence < (x_beta,y_beta):beta<gamma > determines a relation pi subseteq AxB. Player exists wins this round of the game if pi is a partial isomorphism. Otherwise for all wins. The game EFG_gamma^delta (A,B) is defined similarly except that the players play sequences of length<delta at a time. Theorem 1: The following statements are equiconsistent relative to ZFC: (A) There is a weakly compact cardinal. (B) CH and EF_{omega_1}(A,B) is determined for all models A,B of cardinality aleph_2 . Theorem 2: Assume that 2^omega <2^{omega_3} and T is a countable complete first order theory. Suppose that one of (i)-(iii) below holds. Then there are A,B models T of power omega_3 such that for all cardinals 1<theta<=omega_3, EF^theta_{omega_1}(A,B) is non-determined. [(i)] T is unstable. [(ii)] T is superstable with DOP or OTOP. [(iii)] T is stable and unsuperstable and 2^omega <= omega_3. | |
| dc.identifier | https://arxiv.org/abs/math/0212234 | |
| dc.identifier | http://arxiv.org/abs/math/0212234 | |
| dc.identifier | Fund. Math. 175 No. 1 (2002) 79--96 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66017 | |
| dc.subject | Logic | |
| dc.title | More on the Ehrenfencht-Fraisse game of length omega_1 | |
| dc.type | text |