potter set theory and its philosophy pdf

Potter Set Theory And Its Philosophy Pdf

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An approach to the foundations of mathematics that is of relatively recent origin, Scott—Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott and the philosopher George Boolos. Potter , clarified and simplified the approach of Scott , and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers , Peano arithmetic and the other usual number systems , and the theory of relations. This section and the next follow Part I of Potter closely. The background logic is first-order logic with identity. The ontology includes urelements as well as sets , which makes it clear that there can be sets of entities defined by first-order theories not based on sets.

Set Theory and its Philosophy: A Critical Introduction

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this also the case of the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way: by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences.

Set theory is a branch of mathematical logic that studies sets , which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the s. After the discovery of paradoxes in naive set theory , such as Russell's paradox , numerous axiom systems were proposed in the early twentieth century, of which the Zermelo—Fraenkel axioms , with or without the axiom of choice , are the best-known. Set theory is commonly employed as a foundational system for mathematics , particularly in the form of Zermelo—Fraenkel set theory with the axiom of choice.

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Scott–Potter set theory

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I want to do a survey of textbooks in set theory. Amazon returns books for the keywords "set theory". A small somewhat random selection with number of references in Google scholar is the following.

This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory.

Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis , the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary although not literally. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. Download to read the full article text. Boolos, G.

Michael Potter: Set Theory and its Philosophy

Its core is a slightly non-standard development of axiomatic set theory, starting with the concept of a collection and working up through the axiom of choice and some simple cardinal arithmetic—enough to understand the statement and significance of the continuum hypothesis, but not enough to appreciate the singular cardinals hypothesis. From a purely technical perspective, three things make Potter's treatment unusual. First, Potter allows urelements in his basic axiomatization.

Philosophy of Mathematics

Please note that ebooks are subject to tax and the final price may vary depending on your country of residence. Alain Badiou's Being and Event continues to impact philosophical investigations into the question of Being.

‘Set Theory and its Philosophy: A Critical Introduction’ by Michael Potter

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9 See, in particular, [Potter, ], p. 10 See [Feferman et al., ], pp. These examples can be multiplied. 12 For a careful reconstruction of the.

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