Singular cardinals and the PCF theory
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چکیده
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منابع مشابه
Pcf Theory
The abbreviation “PCF” stands for “Possible Cofinalities”. PCF theory was invented by Saharon Shelah to prove upper bounds on exponents of singular cardinals. The starting point of PCF theory is in the realization that the usual exponent function is too coarse for measuring the power set of singular cardinals. Consider the cardinal אω, which is the smallest singular cardinal and has countable c...
متن کاملA Connection between Decomposability of Ultrafilters and Possible Cofinalities
We introduce the decomposability spectrum KD = {λ ≥ ω|D is λ-decomposable} of an ultrafilter D, and show that Shelah’s pcf theory influences the possible values KD can take. For example, we show that if a is a set of regular cardinals, μ ∈ pcf a, the ultrafilter D is |a|-complete and KD ⊆ a, then μ ∈ KD. As a consequence, we show that if λ is singular and for some λ < λ KD contains all regular ...
متن کاملA Note on Powers of Singular Strong Limit Cardinals
We show via a simple forcing argument that if κ ≥ א0 is any cardinal such that κ+ω is a strong limit cardinal, then 2κ +ω < κ+ω4 . Our proof makes use of pcf theory applied only at אω and is generalizable to other contexts. One of the most important results in all of set theory is Shelah’s celebrated theorem [2] that if אω is a strong limit cardinal, then 2אω < אω4 . This theorem is proven via ...
متن کاملReview on Todd Eisworth’s Chapter for the Handbook of Set Theory: “successors of Singular Cardinals”
This chapter offers a comprehensive and lucid exposition of the questions and techniques involved in the study of combinatorics of successors of singular cardinals. What is so special about successors of singular cardinals? They are successor cardinals, but are also similar to inaccessible cardinals, in the sense that there is no maximal regular cardinal below them, meaning for instance that th...
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Axiomatics. The formal axiomatic system of ordinary set theory (ZFC). Models of set theory. Absoluteness. Simple independence results. Transfinite recursion. Ranks. Reflection principles. Constructibility. [4] Infinitary combinatorics. Cofinality. Stationary sets. Fodor’s lemma. Solovay’s theorem. Cardinal exponentiation. Beth and Gimel functions. Generalized Continuum Hypothesis. Singular Card...
متن کاملEmbedding Cohen algebras using pcf theory
Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν adds a generic for the Cohen forcing notion on ν. The following question (problem 5.1 in Miller’s list [Mi91]) is attributed to Rene David and Sy Friedman: Does the product of the forcing notions n2 add a generic for the forcing ω+12? We show here that the answer is ye...
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ورودعنوان ژورنال:
- Bulletin of Symbolic Logic
دوره 1 شماره
صفحات -
تاریخ انتشار 1995