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### Weak Cardinality Theorems for First-Order Logic [chapter]

Till Tantau
2003 Lecture Notes in Computer Science
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.  ...  logo Summary Summary The weak cardinality theorems for first-order logic unify the weak cardinality theorems of automata and recursion theory.  ...

### Model Theoretic Characterizations of Large Cardinals Revisited [article]

Will Boney, Stamatis Dimopoulos, Victoria Gitman, Menachem Magidor
2022 arXiv   pre-print
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  ...

### 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]

Juliette Kennedy, Jouko Väänänen
2021 arXiv   pre-print
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?  ...

### Chain Logic and Shelah's Infinitary Logic [article]

Mirna Džamonja, Jouko Väänänen
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  ,  ,  ,  ,  ,  , or for fragments of first order logic  .  ...

### Logicality and Model Classes

Juliette Kennedy, Jouko Väänänen
2021 Bulletin of Symbolic Logic
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 .  ...

### On the structure of categorical abstract elementary classes with amalgamation [article]

Monica M. VanDieren, Sebastien Vasey
2016 arXiv   pre-print
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).  ...

### On a Question of Hamkins and Löwe on the modal logic of collapse forcing [article]

2021 arXiv   pre-print
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  ...

### 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

Jouko Väänänen
2020 Theoria
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.  ...

### Categorical large cardinals and the tension between categoricity and set-theoretic reflection [article]

Joel David Hamkins, Hans Robin Solberg
2022 arXiv   pre-print
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.  ...

### Maximality of logic without identity [article]

Guillermo Badia, Xavier Caicedo, Carles Noguera
2022 arXiv   pre-print
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 .  ...

### 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.  ...

### An exposition of the compactness of L(Q^cf) [article]

Enrique Casanovas, Martin Ziegler
2020 arXiv   pre-print
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.  ...
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