Small $$\mathfrak {u}(\kappa )$$ at singular $$\kappa $$ with compactness at $$\kappa ^{++}$$

Abstract

We study the relationship between non-trivial values of generalized cardinal invariants at an inaccessible cardinal κ and compactness principles at κ ++ . Let TP(λ), ¬wKH(λ), SR(λ) and DSS(λ) denote the tree property, the negation of the weak Kurepa Hypothesis, stationary reflection and the disjoint stationary sequence property, respectively, at a regular cardinal λ.We show that if the existence of a supercompact cardinal κ with a weakly compact cardinal λ above κ is consistent, then the following are consistent as well (where t(κ) and u(κ) are the tower number and the ultrafilter number, respectively):(i) There is an inaccessible cardinal κ such that κ + < t(κ) = u(κ) < 2 κ and SR(κ ++ ) and DSS(κ ++ ) hold, and(ii) There is an inaccessible cardinal κ such that κ + = t(κ) < u(κ) < 2 κ and SR(κ ++ ), DSS(κ ++ ), TP(κ ++ ) and ¬wKH(κ + ) hold.The cardinals u(κ) and 2 κ can have any reasonable values in these models. We obtain these results by combining the forcing construction from [4] due to Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results related to SR(λ), DSS(λ), TP(λ) and ¬wKH(λ). Apart from u(κ) and t(κ) we also compute the values of b(κ), d(κ), s(κ), r(κ), a(κ), cov(Mκ), add(Mκ), non(Mκ), cof(Mκ) which will all be equal to u(κ).In (ii), we compute p(κ) = t(κ) = κ + by observing that the κ +distributive quotient of the Mitchell forcing adds a tower of size κ + .Finally, as a corollary of the construction, we obtain that (i) and (ii) are also true for κ = ω (starting with a weakly compact cardinal in the ground model).

Trees and Stationary Reflection at Double Successors of Regular Cardinals

“…For preserving club stationary reflection at κ ++ in each of these cases, we use a recent result of the third author (see [15]): Fact 6.7. Suppose = κ ++ , κ <κ = κ, and CSR( ) holds.…”

“…We use another result of the third author (see [15]): Fact 6.11. Suppose is a regular cardinal, SR( + ) holds, and Q is -cc.…”

“…However, Neeman’s proof does not seem to generalize to Cohen forcing at an arbitrary regular . We overcame this obstacle by using a preservation theorem by Honzik and the third author (see [15]). This preservation theorem comes with an analog used to prove Theorem 1.7.…”