Characterizing the powerset by a complete (Scott) sentence

• Autorzy: Ioannis Souldatos
• Języki publikacji: EN

Abstrakty

EN

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence $ϕ_ {ℳ}$ if $ϕ_ {ℳ}$ has a model of size κ, but no model of size κ⁺.
The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $ℵ_{β}$ is characterized by a Scott sentence, then $2^{ℵ_{β+β₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 < β₁ < ω₁. So, the answer to the above question is positive, except the case β₁ = 0 which remains open.
As a consequence we derive that if α ≤ β and $ℵ_{β}$ is characterized by a Scott sentence, then $ℵ_{α+α₁}^{ℵ_{β+β₁}}$ is (homogeneously) characterized by a Scott sentence, for all α₁ < ω₁ and 0 < β₁ < ω₁. Hence, depending on the model of ZFC, we see that the class of characterizable and homogeneously characterizable cardinals is much richer than previously known. Several open questions are mentioned at the end.

Kategorie tematyczne

• 03C75: Other infinitary logic
• 03C35: Categoricity and completeness of theories
• 03E10: Ordinal and cardinal numbers
• 03E75: Applications of set theory
• 03C30: Other model constructions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2013
• Tom: 222
• Numer: 2
• Strony: 131-154

Daty

wydano

2013

Twórcy

autor

Ioannis Souldatos

• Mathematics Department, University of Detroit Mercy, 4001 W. McNichols, Detroit, MI 48221, U.S.A.

Identyfikatory

DOI

10.4064/fm222-2-2