Online research in philosophy

The separation, uniformization, and other properties of the Borel and projective hierarchies over hyperfinite sets are investigated and compared to the corresponding properties in classical descriptive set theory. The techniques used in this investigation also provide some results about countably determined sets and functions, as well as an improvement of an earlier theorem of Kunen and Miller.

If θ is any singular cardinal of cofinality ω 1 , we produce a forcing extension in which MA holds below θ but fails at θ. The failure is due to a partial order which splits a gap of size θ in P(ω).

We consider a well-known partial order of Prikry for producing a collapsing function of minimal degree. Assuming MA + ≠ CH, every new real constructs the collapsing map.

T. K. Menas [4, pp. 225-234] introduced a combinatorial property χ (μ) of a measure μ on a supercompact cardinal κ and proved that measures with this property also have the partition property. We prove here that Menas' property is not equivalent to the partition property. We also show that if α is the least cardinal greater than κ such that P κ α bears a measure without the partition property, then α is inaccessible and Π 2 1 -indescribable.