On the weak Freese-Nation property of P(ω)
نویسندگان
چکیده
Continuing [6], [8] and [15], we study the consequences of the weak Freese-Nation property of (P(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń’s diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (P(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models.
منابع مشابه
On the weak Freese-Nation property of complete Boolean algebras
The following results are proved: (a) In a model obtained by adding א2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of 2μ and cof([μ] א0 ,⊆)...
متن کاملF U N D a M E N T a Mathematicae More Set-theory around the Weak Freese–nation Property
We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese–Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for אω , we can find a counter-ex...
متن کامل3 M ay 1 99 6 More set - theory around the weak Freese - Nation property
For a regular κ, a partial ordering P is said to have the κ Freese-Nation property (the κ-FN for short) if there is a mapping (κ-FN mapping) f : P → [P ] such that for any p, q ∈ P if p ≤ q then there is r ∈ f(p) ∩ f(q) such that p ≤ r ≤ q. Freese and Nation [5] used the א0-FN in a characterization of projective lattices and asked if this property alone already characterizes the projectiveness....
متن کاملA Game on Partial Orderings
We study the determinacy of the game Gκ(A) introduced in [FuKoShe] for uncountable regular κ and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that Gκ(A) is undetermined. For the class of linear orders, the existence of such A depends on the size of κ. In particular we obtain a characterization of κ = κ in terms of determinacy of the game ...
متن کاملar X iv : m at h / 95 05 21 2 v 1 [ m at h . L O ] 1 5 M ay 1 99 5 A game on partial orderings
We study the determinacy of the game Gκ(A) introduced in [FuKoShe] for uncountable regular κ and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that Gκ(A) is undetermined. For the class of linear orders, the existence of such A depends on the size of κ. In particular we obtain a characterization of κ = κ in terms of determinacy of the game ...
متن کامل