Relative Categoricity and Abstraction Principles

Walsh, Sean; Ebels-Duggan, Sean C.

doi:10.1017/s1755020315000052

Cited by 9 publications

(24 citation statements)

References 39 publications

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“…We do not require S i [M ] to be the full power-set of M , for compatibility of our results with those of [24]; that is, we work in the non-standard semantics. Likewise we require our models to satisfy comprehension axioms for all formulae in their signature; these axioms are of the form…”

Section: Why Bother a Sleeping Theorem?mentioning

confidence: 99%

“…Thus, while first-order objects are indistinguishable using only L 0 -definable notions, second-order objects are not. It makes sense, then, that we would want to pay special attention to collections of second-order 10 In [24] the principles CC, ISM, and IPM were deployed as consequences of the principle GC, though GC's well-ordering was in the main results only to show that the restrictions to small concepts and to abstracts were required (see [24, equations 4.22-4.23 and following]). In this paper we will use only the cardinal consequences of GC listed above.…”

Section: Bicardinal Equivalence and Permutation Invariancementioning

confidence: 99%

“…Our aim in these sections will not be to argue for or against a particular version of neo-logicism, but instead to let the Main Theorem shed light on some technical questions of interest. Section 6 will address the sorting of so-called "bad companions" by the Main Theorem; while section 7 proves and discusses the extension of the relative categoricity theorem of [24]. For ease of exposition, we put off some of the more tedious or repetitive proofs to appendices, we conclude with these.…”

Section: Introductionmentioning

confidence: 99%

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Abstraction Principles and the Classification of Second-Order Equivalence Relations

Ebels-Duggan¹

2019

Notre Dame J. Formal Logic

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This paper improves two existing theorems of interest to neo-logicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without non-logical vocabulary, then the bicardinal slice of any equivalence class-those equinumerous elements of the equivalence class with equinumerous complements-can have one of only three profiles. The improvements to Fine's theorem allow for an analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan's relative categoricity theorem. 2 We put aside concerns, like those of Quine in [18], that second-order languages are themselves not "logical" in virtue of their quantifying over higher-order objects.3 This view is associated with Tarski's identification of logical notions as those that are permutation invariant in this sense (see [22]), for a more recent defense see the work of Gila Sher (see [21]). Other relevant discussions can be found in [2] and [16]. The so-called Tarski-Sher thesis identifying logicality with permutation invariance is controversial; that permutation invariance is a necessary feature of logical notions is not (see [15]).

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Section: Why Bother a Sleeping Theorem?mentioning

confidence: 99%

Section: Bicardinal Equivalence and Permutation Invariancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Abstraction Principles and the Classification of Second-Order Equivalence Relations

Ebels-Duggan¹

2019

Notre Dame J. Formal Logic

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“…57-58). 3 It is worth reiterating the remark of (Walsh and Ebels-Duggan 2015) that Pairing is a consequence of GC. It is also worth the separate remark that in ZF set theory, a version of Pairing implies the (set theoretic) Axiom of Choice (see Jech 1973, Theorem 11.7).…”

mentioning

confidence: 98%

“…The further question arises because satisfiability (having a standard model) and consistency (not proving a contradiction) are not the same in second‐order logic. The question was partially answered by Walsh and Ebels‐Duggan (, pp. 21–22); the present note moves us further, but not fully, toward a complete answer.…”

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confidence: 99%

The Nuisance Principle in Infinite Settings

Ebels-Duggan¹

2015

Thought: A Journal of Philosophy

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Neo‐Fregeans have been troubled by the Nuisance Principle (NP), an abstraction principle that is consistent but not jointly (second‐order) satisfiable with the favored abstraction principle HP. We show that logically this situation persists if one looks at joint (second‐order) consistency rather than satisfiability: under a modest assumption about infinite concepts, NP is also inconsistent with HP.

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Asymptotic Quasi-completeness and ZFC

Džamonja

Panza

2018

Trends in Logic

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The axioms ZFC of first order set theory are one of the best and most widely accepted, if not perfect, foundations used in mathematics. Just as the axioms of first order Peano Arithmetic, ZFC axioms form a recursively enumerable list of axioms, and are, then, subject to Gödel's Incompleteness Theorems. Hence, if they are assumed to be consistent, they are necessarily incomplete. This can be witnessed by various concrete statements, including the celebrated Continuum Hypothesis CH. The independence results about the infinite cardinals are so abundant that it often appears that ZFC can basically prove very little about such cardinals. However, we put forward a thesis that ZFC is actually very powerful at some infinite cardinals, but not at all of them. We have to move away from the first few and to look at limits of uncountable cardinals, such as ℵω. Specifically, we work with singular cardinals (which are necessarily limits) and we illustrate that at such cardinals there is a very serious limit to independence and that many statements which are known to be independent on regular cardinals become provable or refutable by ZFC at singulars. In a certain sense, which we explain, the behavior of the set-theoretic universe is asymptotically determined at singular cardinals by the behavior that the universe assumes at the smaller regular cardinals. Foundationally, ZFC provides an asymptotically univocal image of the universe of sets around the singular cardinals. We also give a philosophical view accounting for the relevance of these claims in a platonistic perspective which is different from traditional mathematical platonism.

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Relative Categoricity and Abstraction Principles

Cited by 9 publications

References 39 publications

Abstraction Principles and the Classification of Second-Order Equivalence Relations

Abstraction Principles and the Classification of Second-Order Equivalence Relations

The Nuisance Principle in Infinite Settings

Asymptotic Quasi-completeness and ZFC

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