Uncountable Sets and an Infinite Real Number Game

Uncountable sets and an infinite real number game

The game. Alice and Bob decide to play the following infinite game on the real number line. A subset S of the unit interval [0, 1] is fixed, and then Alice and Bob alternate playing real numbers. Alice moves first, choosing any real number a1 strictly between 0 and 1. Bob then chooses any real number b1 strictly between a1 and 1. On each subsequent turn, the players must choose a point strictly...

Cohesive Sets: Countable and Uncountable

We show that many uncountable admissible ordinals (including some cardinals) as well as all countable admissible ordinals have cohesive subsets. Exactly which cardinals have cohesive subsets, however, is shown to depend on set-theoretic assumptions such as V=L or a large cardinal axiom. The study of recursion theory on the ordinals was initiated by Takeuti and then generalized by several others...

From Finite to Uncountable Sets

Generated topology on infinite sets by ultrafilters

Let $X$ be an infinite set, equipped with a topology $tau$. In this paper we studied the relationship between $tau$, and ultrafilters on $X$. We can discovered, among other thing, some relations of the Robinson's compactness theorem, continuity and the separation axioms. It is important also, aspects of communication between mathematical concepts.

Uncountable γ - sets under axiom CPA gamecube

In the paper we formulate a Covering Property Axiom CPA cube , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in R (which are strongly meager) as well as uncountable γ-sets in R which are not strongly meager. These sets must be of cardinality ω1 < c, since every γ-set is universally null, while CPA cube implies that every unive...