We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ ≥ κ+ if there is an embedding j : V → M with critical point κ such that H(μ)V is included in M and j(κ) > μ. Such j is called a witnessing embedding. Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then th...

statement of problem: dental caries is one of the common diseases in the world. it is an infective disease which is caused by loss of minerals and destruction of dental tissues. one of the most effective ways to diagnose the interproximal caries is radiography. bitewing radiography is the best way for caries diagnosis. purpose: the aim of this study was to evaluate the prevalence of the interpr...

ion. LEQ-λ Γ, X :pκ ` F ≤ F ′ : κ′ Γ ` λXF ≤ λXF ′ : pκ→ κ′ Application. There are two kinds of congruence rules for application: one kind states that if functions F and F ′ are in the subtyping relation, so are their values F G and F ′G at a certain argument G. LEQ-FUN Γ ` F ≤ F ′ : pκ→ κ′ p−1Γ ` G : κ Γ ` F G ≤ F ′G : κ′ The other kind of rules concern the opposite case: If F is a function an...

We characterize the (κ, λ,< μ)-distributive law in Boolean algebras in terms of cut and choose games G <μ(λ), when μ ≤ κ ≤ λ and κ <κ = κ. This builds on previous work to yield game-theoretic characterizations of distributive laws for almost all triples of cardinals κ, λ, μ with μ ≤ λ, under GCH. In the case when μ ≤ κ ≤ λ and κ = κ, we show that it is necessary to consider whether the κ-statio...

We provide a lower bound for the consistency strength of the hypothesis proposed by S. Kamo: ∃κ > א0 s(κ) ≥ κ. 0. Introduction. In [3] ,[10] cardinal invariants on ω are generalized for uncountable regular cardinals and it is shown that some of their properties still hold true for their uncountable counterparts.However,the role of the splitting number s(κ) (for the definition see below) changes...

We study several cardinal, and ordinal–valued functions that are relatives of Hanf numbers. Let κ be an infinite cardinal, and let T ⊆ Lκ+,ω be a theory of cardinality ≤ κ, and let γ be an ordinal ≥ κ . For example we look at 1. μ∗T (γ, κ) := min{μ ∗ : ∀φ ∈ L∞,ω, with rk(φ) < γ, if T has the (φ, μ∗)-order property then there exists a formula φ ′ (x; y) ∈ Lκ+,ω, such that for every χ ≥ κ, T has ...

We present a technique for destroying stationary subsets of Pκκ using partial square sequences. We combine this method with Gitik’s poset for changing the cofinality of a cardinal without adding bounded sets to prove a variety of consistency results concerning saturated ideals and the set S(κ, κ+). In this paper we continue our study of consistency results concerning the set S(κ, κ) = {a ∈ Pκκ ...

Every set of ordinals of cardinality κ in a Prikry extension with a measure over κ contains an old set of arbitrary large cardinality below κ, and, actually, it can be split into countably many old sets. What about sets bigger cardinalities? Clearly, any set of ordinals in a forcing extension of a regular cardinality above the cardinality of the forcing used, contains an old set of the same car...

The games G 1 (κ) and G η <λ(κ) are played by two players in η +complete and max(η+ , λ)-complete Boolean algebras, respectively. For cardinals η, κ such that κ<η = η or κ<η = κ, the (η, κ)-distributive law holds in a Boolean algebra B iff Player 1 does not have a winning strategy in G 1 (κ). Furthermore, for all cardinals κ, the (η,∞)-distributive law holds in B iff Player 1 does not have a wi...