Uncountable γ-sets under axiom CPAcubegame

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...

UNCOUNTABLE INTERSECTIONS OF OPEN SETS UNDER CPAprism

We prove that the Covering Property Axiom CPAprism, which holds in the iterated perfect set model, implies the following facts. • If G is an intersection of ω1-many open sets of a Polish space and G has cardinality continuum, then G contains a perfect set. • There exists a subset G of the Cantor set which is an intersection of ω1-many open sets but is not a union of ω1-many closed sets. The exa...

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

The Axiom for Connected Sets

sists of the set of all ra such that EdneT. The Gödel number of Ed is, of course, d and the Gödel number of ra (i.e. of a string of l's of length w) is 2" —1. Thus A consists of all ra such that d * (2n— 1)G7V We add to (S2) : Axiom. PI —11. Production. Px — y—>Pxl — yy. P represents the set of all ordered pairs (i, j) such that 2'=j. Then we add: Production. Px — yl, Cd — y — z, Toz—^Qx. In th...