On preimages of ultrafilters in ZF

Abstract

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Spectra of uniformity

“…It is not hard to verify, however, that ifẋ ∈ HS and 1 HSẋ ∈Ȧ, then {n | 1 ẋ =ȧn} is finite, and by an easy density argument, 1 does not decide the value of0 ∈ȧn for any finitely many reals. 4 This leads to the sometimes additional requirement that κ is uncountable in the definitions of measurable and strongly compact cardinals. For our purposes, however, it is better to allow ℵ 0 to be considered as measurable or strongly compact.…”

“…In a recent paper [4] by Horst Herrlich, Paul Howard, and Eleftherios Tachtsis, the authors point out that it is open whether or not it is possible that there are ultrafilters on ω 1 , but there are no uniform ultrafilters on ω 1 . This can be done using a slight generalization of Feferman's argument from [3], 5 as we will show in this section.…”

Spectra of uniformity

“…It is not hard to verify, however, that if ẋ ∈ HS and 1 HS ẋ ∈ Ȧ, then {n | 1 ẋ = ȧn} is finite, and by an easy density argument, 1 does not decide the value of 0 ∈ ȧn for any finitely many reals. 4 This leads to the sometimes additional requirement that κ is uncountable in the definitions of measurable and strongly compact cardinals. For our purposes, however, it is better to allow ℵ 0 to be considered as measurable or strongly compact.…”

“…In a recent paper [4] by Horst Herrlich, Paul Howard, and Eleftherios Tachtsis, the authors point out that it is open whether or not it is possible that there are ultrafilters on ω 1 , but there are no uniform ultrafilters on ω 1 . This can be done using a slight generalization of Feferman's argument from [3], 5 as we will show in this section.…”