For any given uncountable cardinal κ with κ = κ, we present a forcing that is <κ-directed closed, has the κ-cc and introduces a lightface definable well-order of H(κ). We use this to define a global iteration that adds such a well-order for all such κ simultaneously and is capable of preserving the existence of many large cardinals in the universe. 2010 Mathematics Subject Classification: 03E47...

On every set A there is a rigid binary relation i.e. such a relation R ⊆ A × A that there is no homomorphism < A,R >→< A,R > except the identity (Vopěnka et al. [1965]). We prove that for each infinite cardinal number κ if card A ≤ 2 then there exists a relation R ⊆ A×A with the following property ∀x ∈ A ∃ {x}⊆A(x)⊆A cardA(x)≤κ ∀ f :A(x)→A f 6=idA(x) f is not a homomorphism of R which implies t...

Assume G.C.H. and κ is the first uncountable cardinal such that there is a non-free κ-free abelian Whitehead group of cardinality κ. We prove that if all κ-free Abelian group of cardinality κ are Whitehead then κ is necessarily an inaccessible cardinal.

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: (a) If m ≥ 1 and the ultrafilter D is (im(λ),im(λ))regular then D is κ-decomposable for some κ with λ ≤ κ ≤ 2 (Theorem 4.3(a)). (b) If λ is a strong limit cardinal and D is (im(λ ),im(λ ))regular then either D is (cf λ, cf λ)-regular or there are arbitrarily large κ < λ for which D is κ-decom...

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ: For weakly compact κ there is no strongest extension of Lκω with the (κ, κ)compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of Lκκ with the (κ, κ)-compactness property and the LöwenheimS...

A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with o(κ) = κ+ 1. On the other hand, it is shown that the strength of above is at least o(κ) = κ. Suppose that κ is an inaccessible cardinal. We wish to find a generic extension (usua...

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ+ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which A = ∅. The first of these cont...

We study several cardinal, and ordinal–valued functions that are relatives of Hanf numbers. Let κ be an infinite cardinality, and let T ⊆ Lκ+,ω be a theory of cardinality ≤ κ, and let γ be an ordinal ≥ κ. Consider (1) μT (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 the (φ , χ)-...

In this study, by generalizing the notion of fuzzy bi-ideals ordered semirings, (∈,∈∨(κ*,qκ))-fuzzy is established. We prove that are but converse not true, and an example provided to support proof. A condition given under which semirings coincide with bi-ideals. An equivalent certain correspondences between presented. Moreover, (κ*,κ)-lower part described depicted in terms several classes semi...

We present a strongly compact version of the Supercompact Magidor forcing ([3]). A variation of it is used to show that the following is consistent: V ⊇ W are transitive models of ZFC+GCH with the same ordinals such that: 1. κ is an inaccessible in W , 2. κ changes its cofinality to ω1 in V witnessed by a club ⟨κα | α < ω1⟩, 3. for every α < ω1, (κ ++ α ) W < κα , 4. (κ++)W = κ+. 1 Preliminary ...