Definable sets of generators in maximal cofinitary groups
نویسندگان
چکیده
A group G Sym(N) is cofinitary if g has finitely many fixed points for every g ∈G except the identity element. In this paper, we discuss the definability of maximal cofinitary groups and some related structures. More precisely, we show the following two results: (1) Assuming V = L, there is a Π1 1 set of permutations on N which generates a maximal cofinitary group. (2) Assuming V = L, there is a Π1 1 mad permutation family in Sym(N). © 2007 Elsevier Inc. All rights reserved. MSC: 03E15; 28A05; 54H05; 20B35; 20B07
منابع مشابه
Definable Maximal Cofinitary Groups
Using countable support iteration of S-proper posets, for some appropriate stationary set S, we obtain a generic extension of the constructible universe, in which b = c = א2 and there is a maximal cofinitary group with a Π2-definable set of generators.
متن کاملA Co-analytic Cohen-Indestructible Maximal cofinitary Group
Assuming that every set is constructible, we find a Π1 maximal cofinitary group of permutations of N which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a Π1 maximal cofinitary group in L.
متن کاملCofinitary Groups, almost Disjoint and Dominating Families
In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(co) is strictly greater than the cardinal numbers D and a. ?
متن کاملTemplate iterations and maximal cofinitary groups
In [2], Jörg Brendle used Hechler’s forcing notion for adding a maximal almost family along an appropriate template forcing construction to show that a (the minimal size of a maximal almost disjoint family) can be of countable cofinality. The main result of the present paper is that ag, the minimal size of maximal cofinitary group, can be of countable cofinality. To prove this we define a natur...
متن کاملMaximal cofinitary groups revisited
Let κ be an arbitrary regular infinite cardinal and let C denote the set of κ-maximal cofinitary groups. We show that if GCH holds and C is a closed set of cardinals such that 1. κ ∈ C, ∀ν ∈ C(ν ≥ κ), 2. if |C| ≥ κ then [κ, |C|] ⊆ C, 3. ∀ν ∈ C(cof(ν) ≤ κ→ ν ∈ C), then there is a generic extension in which cofinalities have not been changed and such that C = {|G| : G ∈ C}. The theorem generalize...
متن کامل