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Weak Cardinality Theorems for First-Order Logic
[chapter]
2003
Lecture Notes in Computer Science
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The 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. ...
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Summary Summary The weak cardinality theorems for first-order logic unify the weak cardinality theorems of automata and recursion theory. ...
doi:10.1007/978-3-540-45077-1_37
fatcat:4jotgwf3nrhkjltwvdldlyanli
Model Theoretic Characterizations of Large Cardinals Revisited
[article]
2022
arXiv
pre-print
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We continue this work, by establishing such characterizations 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 ...
arXiv:2202.00549v1
fatcat:igong7m4nrfkbmbmw2zwzhjhfy
Page 3759 of Mathematical Reviews Vol. , Issue 80J
[page]
1980
Mathematical Reviews
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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
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We suggest that a 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? ...
arXiv:2106.13506v2
fatcat:2kvfqj5dnjgj7f2cs6vermoznm
Chain Logic and Shelah's Infinitary Logic
[article]
2021
arXiv
pre-print
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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] . ...
arXiv:1908.01177v4
fatcat:adduhdtu2zghvoe7llkbr63lli
Logicality and Model Classes
2021
Bulletin of Symbolic Logic
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We suggest that a 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 . ...
doi:10.1017/bsl.2021.42
fatcat:rkhmk3za7bfzfg7ilxjbzpjkeu
On the structure of categorical abstract elementary classes with amalgamation
[article]
2016
arXiv
pre-print
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Our main tool is the symmetry property of splitting (previously isolated by the 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). ...
arXiv:1509.01488v3
fatcat:6f3ff6dn6zdzjntzkasf4ahzpa
On a Question of Hamkins and Löwe on the modal logic of collapse forcing
[article]
2021
arXiv
pre-print
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2020 pre-print arXiv:1609.02633v7 fatcat:tfa4djjtc5bfbnyt3hei2mhdc4Other Versions
We give equiconsistency results 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 ...
arXiv:1609.02633v8
fatcat:h6qsxfxd7rfd5jzsht4ile3qba
Page 6518 of Mathematical Reviews Vol. , Issue 93m
[page]
1993
Mathematical Reviews
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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
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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
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The 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. ...
doi:10.1111/theo.12238
fatcat:xnglhrzxpjcdxfhfjuuw6smseu
Categorical large cardinals and the tension between categoricity and set-theoretic reflection
[article]
2022
arXiv
pre-print
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addition to ZFC_2 either of a 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. ...
arXiv:2009.07164v2
fatcat:crfpzugnlfep7m4ic7nzpz5vca
Maximality of logic without identity
[article]
2022
arXiv
pre-print
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Lindström 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 . ...
arXiv:2203.08722v1
fatcat:s5j6op4r6bcnznmpdcez3brr5i
Page 6616 of Mathematical Reviews Vol. , Issue 92m
[page]
1992
Mathematical Reviews
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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. ...
An exposition of the compactness of L(Q^cf)
[article]
2020
arXiv
pre-print
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We give an exposition of the compactness of L(Q^cf), 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. ...
arXiv:1903.00579v3
fatcat:wdriwx666bdfzd3m5akedrcewm
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