Some remarks on a property of topological cardinal functions

There exists a natural mapping from the smallest transfinite ordinal number , ω, to the corresponding cardinal set , Aleph-0 ,which consists in dissolving the total ordering structure that defines it. One can , in general, state that the cardinal number of any lattice can be obtained through this process. There is however, as this paper claims, no legitimate way to go from Aleph-0 to ω. Granting that a counting process can be applied to through the Axiom of Choice, yet there is no way to know, in advance, which countable transfinite ordinal one will end up with. From this we are led to the definition of a weak transfinite, sigma, whose power set is Aleph-0 . Models for this procedure are developed from the theory of Hilbert Spaces.

1992

We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH: THEOREM (i) If there is a singular strong limit cardinal κ such that 2^κ > kappa^+ then there is an inner model with a cardinal κ such that for all ordinals α<κ there is an ordinal ν < κ with o(ν) > α. (ii) If there is a singular strong limit cardinal κ of uncountable cofinality such that 2^κ > κ^+ then there is an inner model with o(κ) = κ^++. Since this paper was originally submitted, Gitik has improved this result to give exact lower bounds.

Proceedings of the American Mathematical Society, 1974

It is proved that supercompact cardinals can be characterized by combinatorial properties which are generalizations of ineffability. 0. Introduction. A A B is the symmetric difference of A and B. Greek letters will denote ordinals. Pk(A) is the set of all nonempty subsets of A of cardinality less than k. The cardinality of A is \A\. [A]' is the set of all subsets of A of cardinality a. An ultrafilter U on Pk(A) is normal if (a) U is k complete ; (b) for every a e A {P\P e Pk(A), aePjeU; (c) any choice function on Pk(A) is almost everywhere constant. The definition of supercompact cardinal is due to Solovay [6]. A cardinal k is A supercompact if there is a normal ultrafilter on Pk(A). k is supercompact if it is A supercompact for all A. R(a.) is the set of all sets of rank < a. The purpose of this paper is to show that being a supercompact cardinal can be characterized by partition properties which are natural generalizations of those defining ineffable cardinal (Jensen-Kunen [3]). By this we partially settle problem 4 of T. J. Jech [2]. We assume the reader knows the definition of closed subset of an ordinal, unbounded subset of an ordinal, and stationary set, as well as the basic properties of such sets (cf. Fodor [1]). A cardinal k is ineffable iff for any sequence {Ax}x<k, such that Ax^a. for all <t<k, there is A^k and {a|oc</c, An<x=Ax} is stationary in k. An equivalent definition is the following partition property : For any function/: [fc]2-»-2 there is a homogeneous stationary set for/ (that is, a stationary set A such that |/"[/l]2| = l). An ineffable cardinal is weakly compact and therefore inaccessible. 1. Basic facts. A natural generalization of closed, unbounded and stationary sets are the following:

Annals of Mathematical Logic, 1976

It is proved that ff strongly compact cardinals ale consistent, then it is consistent that the fb~t such cardinal is the first measurable. On the othat hand, if it is consistent to asmlne the existence of supcrcompact cardinal, then it is consistent to assume that it is the t~trst strongly compact cardinal. * The auth0t ~shes to e×l~ress his thanks to Miss lrit Masidot, without her help and coop~ation this paper would not have been written. The author is also indebted to the referee for his ve~'y thorough reading of the manuscript and correcting many cnors.

2011

It is proved that there are compact Hausdorff spaces of any pseudoradial order up to ω0 included.

1971

Russian Mathematical Surveys, 2016

The Bulletin of Symbolic Logic, 2002

We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (Σ 0 2 and, under suitable restrictions, Π 0 2) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) Π 0 2-characteristics behave quite chaotically and no new characteristics appear at higher complexity levels. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified proof of Shelah's theorem that the dominating number is less than or equal to the independence number.

2012

Order, 1994

We prove embedding theorems for a~-sequences of ordinals below certain ordinals of countable cofinality. Classifications (1991). 03E05, 04A20, 06A07.