On Strong Lifting Compactness for the Weak$^*$ Topology

On Box, Weak Box and Strong Compactness

One of the most important goals of set theorists over the last few years has been to re-prove old results which previously had used very strong assumptions from hypotheses which, at least prima facie, are weaker. Examples of these abound, including, but certainly not limited to, the work of Woodin and Cummings (see [3]) on the Singular Cardinals Problem, in which results previously obtained by ...

On the Strength of Weak Compactness

We study the logical and computational strength of weak compactness in the separable Hilbert space `2. Let weak-BW be the statement the every bounded sequence in `2 has a weak cluster point. It is known that weak-BW is equivalent to ACA0 over RCA0 and thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstraß principle BW. We show that weak-BW is instance-wise equivalent to Π2-...

On the Strong Equality between Supercompactness and Strong Compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V |= ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V [G] |= ZFC + GCH in which, (a) (preservation) for κ ≤ λ regular, if V |...

Sequential Compactness for the Weak Topology of Vector Measures in Certain Nuclear Spaces

We give a sequential compactness criterion for the weak topology of vector measures with values in certain nuclear spaces, such as the space S of all rapidly decreasing, infinitely differentiable functions, the space D of all test functions, and the strong duals of those spaces. This result contains Prokhorov–LeCam’s criterion for real measures and applies to cases which are not covered by März...

Weak Compactness of the Set of ε-Extensions