Magidor–Malitz reflection

Abstract

In this paper we investigate the consistency and consequences of the downward Löwenheim-Skolem-Tarski theorem for extension of the first order logic by the Magidor-Malitz quantifier. We derive some combinatorial results and improve the known upper bound for the consistency of Chang's Conjecture at successor of singular cardinals.

Usuba's principle $UB_λ$ can fail at singular cardinals

“…In the next section we show that UB ℵω implies the failure of (ℵ ω+1,ℵω ) ։ (ℵ 1 , ℵ 0 ) as well, and hence by the results of [3] (see also [1] and [2], where the consistency of (ℵ ω+1,ℵω ) ։ (ℵ 1 , ℵ 0 ) is proved using weaker large cardinal assumptions) UB ℵω can fail. We also need the following notion.…”

“…It is consistent, relative to the existence of large cardinals, that UB ℵω fails. 2 this assumption is used to guarantee clause (3) in definition of f holds.…”

Stationary and Closed Rainbow subsets

“…Here is one way to demonstrate the hypothesis of Theorem 2.8 is consistent relative to the existence of large cardinals. The argument is essentially due to Levinski, Magidor and Shelah [10], while the large cardinal hypothesis needed is improved by Hayut [7]. By [10] and [7], with the appropriate large cardinal hypothesis (for example, the existence of an (ω + 1)-subcompact cardinal), we may suppose in the ground model, GCH holds and there are two strongly inaccessible cardinals κ < λ satisfying (λ +ω+1 , λ +ω ) ։ (stat(κ +ω+1 ∩ cof(ω)), κ +ω ).…”

“…The argument is essentially due to Levinski, Magidor and Shelah [10], while the large cardinal hypothesis needed is improved by Hayut [7]. By [10] and [7], with the appropriate large cardinal hypothesis (for example, the existence of an (ω + 1)-subcompact cardinal), we may suppose in the ground model, GCH holds and there are two strongly inaccessible cardinals κ < λ satisfying (λ +ω+1 , λ +ω ) ։ (stat(κ +ω+1 ∩ cof(ω)), κ +ω ). More precisely, the symbol means for any regular µ ≥ λ +ω+1 and any countable language L, there exists M ≺ (H(µ), L) such that |M ∩ λ +ω+1 | = κ +ω+1 and |M ∩ λ +ω | = κ +ω and in addition, M ∩ λ +ω+1 ∩ cof (ω) contains a set A of order type κ +ω+1 stationary in its supremum.…”

The generalized Kurepa hypothesis at singular cardinals

“…In particular, F is not a KH λ -family. See also [3], where the large cardinal assumption is reduced to the existence of a (+ω + 1)subcompact cardinal κ.…”