The weak ultrafilter axiom

The main theorem is that the Ultrafilter Axiom of Woodin (J Math Log 11(2):115–37, 2011) must fail at all cardinals where the Axiom I0 holds, in all non-strategic extender models subject only to fairly general requirements on the non-strategic extender model.MathematicsPhilosoph

Social Welfare Functions for Economic Environments with and without the Pareto Principle

Social welfare functions for private goods economies with classical preferences are considered. It is shown that every social welfare function satisfying a weak nonimposition condition and the independence of irrelevant alternatives axiom is of one of the following forms. It is either null or the class of decisive coalitions is an ultrafilter or the class of anti-decisive coalitions is an ultrafilter

Non-principal ultrafilters, program extraction and higher order reverse mathematics

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. Let U be the statement that a non-principal ultrafilter exists and let ACA_0^{\omega} be the higher order extension of ACA_0. We show that ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that ACA_0^{\omega}+\U is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U a realizing term in G\"odel's system T can be extracted. This means that one can extract a term t, such that A(f,t(f))