The Nuisance Principle in Infinite Settings

Ebels-Duggan, Sean C.

doi:10.1002/tht3.186

Cited by 5 publications

(1 citation statement)

References 12 publications

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“…11 Notable bad companions include Boolos's (1990) and Wright's (1997) parity and nuisance principles and Weir's (2003) distraction principles. Ebels-Duggan (2015) shows that, given a modest assumption, the nuisance principle and HP are inconsistent (and not merely unsatisfiable). 12 See, for instance, Weir (2003) for discussion of various early proposals, and Cook and Linnebo (2018) for a classification of some more recent attempts.…”

Section: Neo-logicist Abstractionismmentioning

confidence: 99%

Linnebo's Abstractionism and the Bad Company Problem

Studd

2021

Theoria

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In Thin Objects: An Abstractionist Account, Linnebo offers what he describes as a "simple and definitive" solution to the bad company problem facing abstractionist accounts of mathematics. "Bad" abstraction principles can be rendered "good" by taking abstraction to have a predicative character. But the resulting predicative axioms are too weak to recover substantial portions of mathematics. Linnebo pursues two quite different strategies to overcome this weakness in the case of set theory and arithmetic. I argue that neither infinitely iterated abstraction nor abstraction on possible specifications fully resolves the bad company problem.

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Section: Neo-logicist Abstractionismmentioning

confidence: 99%

Linnebo's Abstractionism and the Bad Company Problem

Studd

2021

Theoria

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What Russell Should Have Said to Burali–forti

Florio¹,

Leach-Krouse

2017

The Review of Symbolic Logic

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The paradox that appears under Burali–Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be–absurdly–an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali–Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali–Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic.

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Deductive Cardinality Results and Nuisance-Like Principles

Ebels-Duggan¹

2021

The Review of Symbolic Logic

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The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof.

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The Nuisance Principle in Infinite Settings

Cited by 5 publications

References 12 publications

Linnebo's Abstractionism and the Bad Company Problem

Linnebo's Abstractionism and the Bad Company Problem

What Russell Should Have Said to Burali–forti

Deductive Cardinality Results and Nuisance-Like Principles

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