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Switch to: References
Citations of:
Recent Results in Set Theory
Journal of Symbolic Logic 37 (4):765-766 (1972)
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The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although Kreisel's conclusion (...) |
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This paper is concerned with the way different axiom systems for set theory can be justified by appeal to such intuitions as limitation of size, predicativity, stratification, etc. While none of the different conceptions historically resulting from the impetus to provide a solution to the paradoxes turns out to rest on an intuition providing an unshakeable foundation,'each supplies a picture of the set-theoretic universe that is both useful and internally well motivated. The same is true of more recently proposed axiom (...) |
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This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory. |
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Scientific knowledge develops in an increasingly fragmentary way.A multitude of scientific disciplines branch out. Curiosity for thisdevelopment leads into quests for a unifying understanding. To a certain extent, foundational studies provide such unification. There is a tendency, however, also of a fragmentary growth of foundational studies, like in a multitude of disciplinaryfoundations. We suggest to look at the foundational problem, not primarily as a search for foundations for one discipline in another, as in some reductionist approach, but as a steady (...) |
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James Walmsley in “Categoricity and Indefinite Extensibility” argues that a realist about some branch of mathematics X (e.g. arithmetic) apparently cannot use the categoricity of an axiomatisation of X to justify her belief that every sentence of the language of X has a truth-value. My discussion note first corrects Walmsley’s formulation of his claim. It then shows that his argument for it hinges on the implausible idea that grasping that there is some model of the axioms amounts to grasping that (...) |
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Pour le philosophe intéressé aux structures et aux fondements du savoir théorétique, à la constitution d'une « méta-théorétique «, θεωρíα., qui, mieux que les « Wissenschaftslehre » fichtéenne ou husserlienne et par-delà les débris de la métaphysique, veut dans une intention nouvelle faire la synthèse du « théorétique », la logique mathématique se révèle un objet privilégié. |
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This article addresses the question of fundamental entities in set theory. It takes up J. Hamkins’ claim that models of set theory are such fundamental entities and investigates it using the methodology of P. Maddy’s naturalism, Second Philosophy. In accordance with this methodology, I investigate the historical case study of the use of models in the introduction of forcing, compare this case to contemporary practice and give a systematic account of how set-theoretic practice can be said to introduce models as (...) |
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Two unpublished contributions to meetings can be found in the Alfred Tarski Papers, at the University of California, Berkeley. The meetings took place in 1965, in Chicago and London, respectively.... |
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