Representability and compactness for pseudopowers

Todd Eisworth

doi:10.1007/s00153-021-00780-9

Home
Search
Representability and compactness for pseudopowers

Open Access
2
https://doi.org/10.1007/s00153-021-00780-9

Representability and compactness for pseudopowers

May 28, 2021
Archive for Mathematical Logic
Todd Eisworth

Listen

Abstract
Literature Map
References
Citations
Similar Papers

Abstract

We prove a compactness theorem for pseudopower operations of the form $${{\,\mathrm{pp}\,}}_{\Gamma (\mu ,\sigma )}(\mu )$$ where $$\aleph _0<\sigma ={{\,\mathrm{cf}\,}}(\sigma )\le {{\,\mathrm{cf}\,}}(\mu )$$ . Our main tool is a result that has Shelah’s cov versus pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set A of regular cardinals for which $${{\,\mathrm{pcf}\,}}(A)$$ has an inaccessible accumulation point.

Similar Papers

Research Article
15
10.2307/2275578

The cofinality spectrum of the infinite symmetric group

Sep 01, 1997
Journal of Symbolic Logic
Saharon Shelah + 1 more +1

Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem.is the main result of this paper.Theorem. Suppose that V ⊨ GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.(a) C contains a maximum element.(b) If μ is an inaccessible cardinal such thatμ = sup(C ∩ μ), thenμ ∈ C.(c) if μ is a singular cardinal such thatμ = sup(C ∩ μ), thenμ+ ∈ C.Then there exists a c.c.c. notion of forcing ℙ such that Vℙ ⊨ CF(S) = C.We shall also investigate the connections between the cofinality spectrum and pcf theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.

Research Article
101
10.1016/0168-0072(90)90057-9

Shelah's pcf theory and its applications

Dec 01, 1990
Annals of Pure and Applied Logic
Maxim R Burke + 1 more +1

Shelah's pcf theory and its applications

Research Article
3
10.1016/0003-4843(81)90015-2

Recursively invariant β-recursion theory

Oct 01, 1981
Annals of Mathematical Logic
Wolfgand Maass

Recursively invariant β-recursion theory