Characterizing the powerset by a complete (Scott) sentence

Ioannis Souldatos. "Characterizing the powerset by a complete (Scott) sentence." Fundamenta Mathematicae 222.2 (2013): 131-154. <http://eudml.org/doc/282820>.

@article{IoannisSouldatos2013,
abstract = {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.},
author = {Ioannis Souldatos},
journal = {Fundamenta Mathematicae},
keywords = {infinitary logic; Scott sentence; complete sentence; characterizable cardinals; powerset},
language = {eng},
number = {2},
pages = {131-154},
title = {Characterizing the powerset by a complete (Scott) sentence},
url = {http://eudml.org/doc/282820},
volume = {222},
year = {2013},
}

TY - JOUR
AU - Ioannis Souldatos
TI - Characterizing the powerset by a complete (Scott) sentence
JO - Fundamenta Mathematicae
PY - 2013
VL - 222
IS - 2
SP - 131
EP - 154
AB - 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.
LA - eng
KW - infinitary logic; Scott sentence; complete sentence; characterizable cardinals; powerset
UR - http://eudml.org/doc/282820
ER -