Singular Cardinals and Square Properties
نویسنده
چکیده
We analyze the effect of singularizing cardinals on square properties. An old theorem of Dzamonja-Shelah/Gitik says that if you singularize an inaccessible cardinal while preserving its successor, then κ,ω holds in the bigger model. We extend this to the situation where a finite interval of cardinals above κ is collapsed. More precisely, we show that if V ⊂ W , κ is inaccessible in V , cf (κ V ) = ω for all 0 ≤ i ≤ n, and κ V = κ + W , then W |= κ,ω.
منابع مشابه
Singular Cardinals and Square Properties Menachem Magidor and Dima Sinapova
We analyze the effect of singularizing cardinals on square properties. By work of Džamonja-Shelah and of Gitik, if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then κ,ω holds in the bigger model. We extend this to the situation where every regular cardinal in an interval [κ, ν] is singularized, for some regular cardinal ν. More precisely, we s...
متن کاملI. Coherent Sequences
I Coherent Sequences 3 by Stevo Todorcevic 1 The Space of Countable Ordinals . . . . . . . . . . . . . . . . . . 4 2 Subadditive Functions . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Steps and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 The Trace and the Square-Bracket Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 A Square-Bracket Ope...
متن کاملMaking All Cardinals Almost Ramsey ∗ † ‡ Arthur
We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ¬ACω in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular c...
متن کاملGuessing models and generalized Laver diamond
We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinals properties can be defined in terms of suitable elementary embeddings j : ...
متن کاملOn the Singular Cardinals
We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen, Square on singular cardinals. A combinatorial principle of great importance in set theory is the Global principle of Jensen [6]: Global : There exists 〈Cα | α a singular ordinal 〉 such that for each α, Cα is a closed unbounded subset of α of ordertype less than α, the li...
متن کامل