Chain logic and Shelah’s infinitary logic

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F21%3A00550973" target="_blank" >RIV/67985840:_____/21:00550973 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s11856-021-2207-0" target="_blank" >https://doi.org/10.1007/s11856-021-2207-0</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11856-021-2207-0" target="_blank" >10.1007/s11856-021-2207-0</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Chain logic and Shelah’s infinitary logic

Popis výsledku v původním jazyce

For a cardinal of the form κ = בκ, Shelah’s logic Lκ1 has a characterisation as the maximal logic above ∪ λ<κLλ,ω satisfying a strengthening of the undefinability of well-order. Karp’s chain logic [20] Lκ, κc is known to satisfy the undefinability of well-order and interpolation. We prove that if κ is singular of countable cofinality, Karp’s chain logic [20] is above Lκ1. Moreover, we show that if κ is a strong limit of singular cardinals of countable cofinality, the chain logic L<κ,<κc∪λ<κLλ,λc is a maximal logic with chain models to satisfy a version of the undefinability of well-order. We then show that the chain logic gives a partial solution to Problem 1.4 from Shelah’s [28], which asked whether for κ singular of countable cofinality there was a logic strictly between Lκ+,ω and Lκ+,κ+ having interpolation. We show that modulo accepting as the upper bound a model class of Lκ, κ, Karp’s chain logic satisfies the required properties. In addition, we show that this chain logic is not κ-compact, a question that we have asked on various occasions. We contribute to further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic Lκ1 in satisfying interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a completeness theorem.

Název v anglickém jazyce

Chain logic and Shelah’s infinitary logic

Popis výsledku anglicky

For a cardinal of the form κ = בκ, Shelah’s logic Lκ1 has a characterisation as the maximal logic above ∪ λ<κLλ,ω satisfying a strengthening of the undefinability of well-order. Karp’s chain logic [20] Lκ, κc is known to satisfy the undefinability of well-order and interpolation. We prove that if κ is singular of countable cofinality, Karp’s chain logic [20] is above Lκ1. Moreover, we show that if κ is a strong limit of singular cardinals of countable cofinality, the chain logic L<κ,<κc∪λ<κLλ,λc is a maximal logic with chain models to satisfy a version of the undefinability of well-order. We then show that the chain logic gives a partial solution to Problem 1.4 from Shelah’s [28], which asked whether for κ singular of countable cofinality there was a logic strictly between Lκ+,ω and Lκ+,κ+ having interpolation. We show that modulo accepting as the upper bound a model class of Lκ, κ, Karp’s chain logic satisfies the required properties. In addition, we show that this chain logic is not κ-compact, a question that we have asked on various occasions. We contribute to further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic Lκ1 in satisfying interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a completeness theorem.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GX20-31529X" target="_blank" >GX20-31529X: Abstraktní konvergenční schémata a jejich složitost</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Israel Journal of Mathematics

ISSN

0021-2172

e-ISSN

1565-8511

Svazek periodika

245

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

IL - Stát Izrael

Počet stran výsledku

42

Strana od-do

93-134

Kód UT WoS článku

000705737700008

EID výsledku v databázi Scopus

2-s2.0-85116478785