Chain logic and Shelah’s infinitary logic

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

Chain logic and Shelah’s infinitary logic

Original language description

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.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GX20-31529X" target="_blank" >GX20-31529X: Abstract convergence schemes and their complexities</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2021

Confidentiality

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

Name of the periodical

Israel Journal of Mathematics

ISSN

0021-2172

e-ISSN

1565-8511

Volume of the periodical

245

Issue of the periodical within the volume

1

Country of publishing house

IL - THE STATE OF ISRAEL

Number of pages

42

Pages from-to

93-134

UT code for WoS article

000705737700008

EID of the result in the Scopus database

2-s2.0-85116478785