On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals

Bagaria, Joan; Väänánen, Jouko

doi:10.1017/jsl.2015.60

Cited by 11 publications

(34 citation statements)

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“…In the final chapter of the thesis we study Löwenheim-Skolem theorems for logics extending first-order logic. We extend the work done in [1] by relating upward Löwenheim-Skolem theorems for strong logics to reflection principles in set theory. Finally, we apply our framework to the study of the large cardinal strength of the upward Löwenheim-Skolem theorem for second-order logic; we provide both upper and lower bounds.…”

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confidence: 99%

The Theory of the Generalised Real Numbers and Other Topics in Logic

Galeotti¹

2019

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The real line R and Baire space are two of the most fundamental objects in descriptive set theory and have been studied extensively. In recent years, set theorists have become increasingly interested in generalised Baire spaces κ κ , i.e., the sets of functions from κ to κ for an uncountable cardinal κ. In the thesis, two κ-analogues of R are discussed, one at https://www.cambridge.org/core/terms.

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confidence: 99%

The Theory of the Generalised Real Numbers and Other Topics in Logic

Galeotti¹

2019

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“…However, Lücke [Lüc21] has established that SR − PwSet is much weaker than SR PwSet . Indeed, he shows 5 In [BV16] it is only assumed that κ witnesses SR − Cd,WC . However, Philipp Lücke [Lüc21] has shown that some additional assumption on κ is needed.…”

Section: Theorem 64 ([Bv16]) If Sr −mentioning

confidence: 99%

Large cardinals as principles of Structural Reflection

Bagaria¹

2021

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After discussing the limitations inherent to all settheoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960's, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflection implicit in Cantor's original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a comprehensive survey of results showing that different forms of the new principles of Structural Reflection are equivalent to well-known large cardinals axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter.

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“…First, recall that classical work of Magidor in [16] yields a characterization of supercompactness through model-theoretic reflection by showing that second-order logic has a Löwenheim-Skolem-Tarski cardinal (see [5,Definition 6.1]) if and only if there exists a supercompact cardinal. Moreover, Magidor's results show that if these equivalent statements hold true, then the least Löwenheim-Skolem-Tarski cardinal for second-order logic is equal to the least supercompact cardinal.…”

Section: Introductionmentioning

confidence: 99%

“…In order to precisely formulate the relevant results from [2], [3] and [5], we first need to discuss certain subclasses of Σ 2 -formulas defined through standard refinements of the Lévy hierarchy of formulas. Given a first-order language L that extends the language L ∈ of set theory, an L-formula is a Σ 0 -formula if it is contained in the smallest class of L-formulas that contains all atomic L-formulas and is closed under negations, conjunctions and bounded existential quantification.…”

Section: Introductionmentioning

confidence: 99%

“…The closure properties of the class of all L ∈ -formulas that are ZFC-provably equivalent to Σ n+1 -formulas then ensure that if R is a class that is Π n -definable in the parameter z, then every class that is Σ n (R)-definable in the parameter z is also Σ n+1 -definable in this parameter. Using Väänänen's notion of symbiosis between model-and set-theoretic reflection principles introduced in [19], Bagaria and Väänänen connected reflection principles for secondorder logic with the principle SR by showing that a cardinal κ is a Löwenheim-Skolem-Tarski cardinal for second-order logic if and only if SR C (κ) holds for every class C of structures that is Σ 1 (P wSet)-definable without parameters (see [5,Theorem 5.5] and [5,Lemma 7.1]), where P wSet denotes the Π 1 -definable class of all pairs of the form x, P(x) . In combination with the results from [16] discussed above, this equivalence provides a characterization of the first supercompact cardinal through the validity of the principle SR C (κ) for all Σ 1 (P wSet)definable classes of structures.…”

Section: Introductionmentioning

confidence: 99%

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Structural reflection, shrewd cardinals and the size of the continuum

Lücke¹

2021

Preprint

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Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle SR − introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle SR − for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor's classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak Π 1 n -indescribability, can be canonically characterized through localized versions of the principle SR − . Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin's weakly compact embedding property is equivalent to Lévy's notion of weak Π 1 1 -indescribability.2020 Mathematics Subject Classification. 03E55, 03C55, 03E47. Key words and phrases. Structural reflection, large cardinals, Σ 2 -definability, elementary embeddings, shrewd cardinals, weakly compact embedding property.The author would like to thank Joan Bagaria for several helpful discussions on the topic of this work and various comments on earlier versions of this paper. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 842082 (Project SAIFIA: Strong Axioms of Infinity -Frameworks, Interactions and Applications).1 Throughout this paper, we will use the term large cardinal notion to refer to properties of cardinals that imply weak inaccessibility.2 In the following, the term structure refers to structures for countable first-order languages. It should be noted that all results cited and proven below remain true if we restrict ourselves to finite languages.

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On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals

Cited by 11 publications

References 15 publications

The Theory of the Generalised Real Numbers and Other Topics in Logic

The Theory of the Generalised Real Numbers and Other Topics in Logic

Large cardinals as principles of Structural Reflection

Structural reflection, shrewd cardinals and the size of the continuum

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