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Weak Cardinality Theorems for First-Order Logic
[chapter]

2003
*
Lecture Notes in Computer Science
*

The

doi:10.1007/978-3-540-45077-1_37
fatcat:4jotgwf3nrhkjltwvdldlyanli
*First**Weak**Cardinality**Theorem**Theorem*Let S be a*logical*structure with universe U and let A ⊆ U. ... The*First**Weak**Cardinality**Theorem**Theorem*Let S be a*logical*structure with universe U and let A ⊆ U. ... logo Summary Summary The*weak**cardinality**theorems**for**first*-*order**logic*unify the*weak**cardinality**theorems*of automata and recursion theory. ...##
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Model Theoretic Characterizations of Large Cardinals Revisited
[article]

2022
*
arXiv
*
pre-print

We continue this work, by establishing such characterizations

arXiv:2202.00549v1
fatcat:igong7m4nrfkbmbmw2zwzhjhfy
*for*Woodin*cardinals*(and variants), various virtual large*cardinals*, and subtle*cardinals*. ... In [Bon20], model theoretic characterizations of several established large*cardinal*notions were given. ...*For*example, ω is the strong compactness*cardinal*of*first*-*order**logic*, a weakly compact*cardinal*κ is a*weak*compactness*cardinal*of the infinitary*logic*L κ,κ , and a strongly compact*cardinal*κ is a ...##
###
Page 3759 of Mathematical Reviews Vol. , Issue 80J
[page]

1980
*
Mathematical Reviews
*

A
similar

*theorem*is proved*for**ordered*fields. ... Therefore the concept of homomor- phism is not relevant*for*geometry. In addition, he shows that*weak*planarity has no*first**order*translation. ...##
###
Logicality and Model Classes
[article]

2021
*
arXiv
*
pre-print

We suggest that a

arXiv:2106.13506v2
fatcat:2kvfqj5dnjgj7f2cs6vermoznm
*logic*is the more*logical*the closer it is to*first**order**logic*. ... We investigate which characteristics of*logics*, such as variants of the Löwenheim-Skolem*Theorem*, Completeness*Theorem*, and absoluteness, are relevant from the*logicality*point of view, continuing earlier ... Sort*Logic*We can ask, is there a*logic*in which every model class whatsoever is definable, irrespective of the*cardinality*of the domain? That is, not just*cardinal*dependently? ...##
###
Chain Logic and Shelah's Infinitary Logic
[article]

2021
*
arXiv
*
pre-print

*For*a

*cardinal*of the form κ=ℶ_κ, Shelah's

*logic*L^1_κ has a characterisation as the maximal

*logic*above ⋃_λ<κ L_λ, ω satisfying Strong Undefinability of Well

*Order*(SUDWO). ... In addition it has a Completeness

*Theorem*. ... versions of modal

*logic*[9] , [34] , [21] , [13] , [14] , [37] , or

*for*fragments of

*first*

*order*

*logic*[35] . ...

##
###
Logicality and Model Classes

2021
*
Bulletin of Symbolic Logic
*

We suggest that a

doi:10.1017/bsl.2021.42
fatcat:rkhmk3za7bfzfg7ilxjbzpjkeu
*logic*is the more*logical*the closer it is to*first**order**logic*. We also offer a refinement of the result of McGee that*logical*properties * ... We investigate which characteristics of*logics*, such as variants of the Löwenheim-Skolem*Theorem*, Completeness*Theorem*, and absoluteness, are relevant from the*logicality*point of view, continuing earlier ... In the case of*first**order**logic*the Löwenheim-Skolem*Theorem*tells us that κ can be taken to be ℵ 0 , so in this sense*first**order**logic*is indifferent to*cardinalities*above ℵ 0 . ...##
###
On the structure of categorical abstract elementary classes with amalgamation
[article]

2016
*
arXiv
*
pre-print

Our main tool is the symmetry property of splitting (previously isolated by the

arXiv:1509.01488v3
fatcat:6f3ff6dn6zdzjntzkasf4ahzpa
*first*author). The key lemma deduces symmetry from failure of the*order*property. ...*Theorem*Let μ>LS (K). ...*For*example, we show how to obtain*weak*tameness (i.e. tameness over saturated models) from categoricity in a big-enough*cardinal*(this is*Theorem*4.14). ...##
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On a Question of Hamkins and Löwe on the modal logic of collapse forcing
[article]

2021
*
arXiv
*
pre-print

We give equiconsistency results

arXiv:1609.02633v8
fatcat:h6qsxfxd7rfd5jzsht4ile3qba
*for*two weaker versions of this property. ... Hamkins and Löwe asked whether there can be a model N of set theory with the property that N≡ N[g] whenever g is a generic collapse of a*cardinal*of N onto ω. ... However we*first*give following result, which both provides the justification*for*our characterization of Hamkins-Löwe on a closed unbounded set as a*weak*Hamkins-Löwe property, and, together with*Theorem*...##
###
Page 6518 of Mathematical Reviews Vol. , Issue 93m
[page]

1993
*
Mathematical Reviews
*

The

*weak*Chang conjecture*for*a successor ordinal A = p’ is the following assertion: Whenever 2 is a*first*-*order*structure with a countable language and A+ C Y, then there is an a <A such that*for*all ... The model spaces of L(Q,) are not normal*for*vocabularies of uncountable power > @,. It also follows that*first*-*order**logic*is the only finite-dependence*logic*...##
###
Page 33 of Mathematical Reviews Vol. , Issue 88a
[page]

1988
*
Mathematical Reviews
*

The author notes that Russell’s and the Q-paradoxes are essentially derivable in

*first*-*order**logic*, without special set-theoretic assump- tions. ...*for*all count- able a. This*theorem*gives the consistency of 2% — (2%°, a)?*for*all a@ < @ relative to the existence of a weakly compact*cardinal*. ...##
###
An Overview of Saharon Shelah's Contributions to Mathematical Logic, in Particular to Model Theory

2020
*
Theoria
*

The

doi:10.1111/theo.12238
fatcat:xnglhrzxpjcdxfhfjuuw6smseu
*first*is in model theory and the other two are in set theory. ... I will give a brief overview of Saharon Shelah's work in mathematical*logic*. I will focus on three transformative contributions Shelah has made: stability theory, proper forcing and PCF theory. ... Acknowledgements This overview is based on a lecture the author gave at the Rolf Schock Prize Symposium in*Logic*and Philosophy in Stockholm, 2018. ...##
###
Categorical large cardinals and the tension between categoricity and set-theoretic reflection
[article]

2022
*
arXiv
*
pre-print

addition to ZFC_2 either of a

arXiv:2009.07164v2
fatcat:crfpzugnlfep7m4ic7nzpz5vca
*first*-*order*sentence, a*first*-*order*theory, a second-*order*sentence or a second-*order*theory. ... Thus we mount an analysis of the categorical large*cardinals*. ... In light of the Löwenheim-Skolem*theorem*, which prevents categoricity*for*infinite structures in*first*-*order**logic*, these categorical theories are generally made in second-*order**logic*. ...##
###
Maximality of logic without identity
[article]

2022
*
arXiv
*
pre-print

Lindström

arXiv:2203.08722v1
fatcat:s5j6op4r6bcnznmpdcez3brr5i
*theorem*obviously fails as a characterization of ℒ_ωω^-,*first*-*order**logic*without identity. ... In this note we provide a fix: we show that ℒ_ωω^- is maximal among abstract*logics*satisfying a*weak*form of the isomorphism property (suitable*for*identity-free languages and studied in ), the Löwenheim–Skolem ... The classical Lindström*theorems*clearly fail*for**first*-*order**logic*without identity (L − ωω ) since*first*-*order**logic*with identity (L ωω ) is a proper extension of L − ωω . ...##
###
Page 6616 of Mathematical Reviews Vol. , Issue 92m
[page]

1992
*
Mathematical Reviews
*

In particular, the following

*theorems*are proved without AC: (1)*For*every*cardinal*number m > 1 the following conditions are equivalent: (i) m? ... Both*theorems*are easy consequences of a more technical*theorem*. G. Asser (Greifswald) 92m:03080 03E25 03E10 04A25 Shannon, Gary P. (1-CASSC) A note on some*weak*forms of the axiom of choice. ...##
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An exposition of the compactness of L(Q^cf)
[article]

2020
*
arXiv
*
pre-print

We give an exposition of the compactness of L(Q^cf),

arXiv:1903.00579v3
fatcat:wdriwx666bdfzd3m5akedrcewm
*for*any set C of regular*cardinals*. ... The assumption only applies to a previous result on a*logic*stronger than*first*-*order**logic*even*for*countable models. ... In*weak*structures every L(Q cf )-formula is equivalent to a*first*-*order*L *formula, and conversely. So the L(Q cf )-model theory of*weak*structures is the same as their*first*-*order*model theory. ...
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