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Set Theory: Boolean-Valued Models and Independence Proofs

John L. Bell

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198568520
eISBN:
9780191717581
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568520.001.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated ... More


Further cardinal arithmetic

Michael Potter

in Set Theory and its Philosophy: A Critical Introduction

Published in print:
2004
Published Online:
September 2011
ISBN:
9780199269730
eISBN:
9780191699443
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199269730.003.0022
Subject:
Philosophy, Logic/Philosophy of Mathematics

The axiom of choice leads to a considerable simplification of the arithmetic of cardinals, but even so it leaves some questions in this domain unsettled. This chapter focuses on these issues. Topics ... More


Realism in Mathematics

Penelope Maddy

Published in print:
1992
Published Online:
November 2003
ISBN:
9780198240358
eISBN:
9780191597978
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/019824035X.001.0001
Subject:
Philosophy, Logic/Philosophy of Mathematics

Many mathematicians understand their work as an effort to describe the denizens and features of an abstract mathematical world or worlds. Most philosophers of mathematics consider views of this sort ... More


Axioms

Penelope Maddy

in Realism in Mathematics

Published in print:
1992
Published Online:
November 2003
ISBN:
9780198240358
eISBN:
9780191597978
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/019824035X.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics

Pursues the theoretical level of the two‐tiered epistemology of set theoretic realism, the level at which more abstract axioms can be justified by their consequences at more intuitive levels. I ... More


Objectivity in Mathematical Knowledge

José Ferreirós

in Mathematical Knowledge and the Interplay of Practices

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691167510
eISBN:
9781400874002
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691167510.003.0009
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new ... More


Naturalism in Mathematics

Penelope Maddy

Published in print:
2000
Published Online:
November 2003
ISBN:
9780198250753
eISBN:
9780191597961
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/0198250754.001.0001
Subject:
Philosophy, Logic/Philosophy of Mathematics

It is well known that certain natural statements of set theory, like Cantor's continuum hypothesis (CH), cannot be proved or disproved on the basis of the standard axioms (Zermelo–Fraenkel with ... More


FORCING AND SOME INDEPENDENCE PROOFS

John L. Bell

in Set Theory: Boolean-Valued Models and Independence Proofs

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198568520
eISBN:
9780191717581
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568520.003.0003
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

The forcing relation between elements of a partially ordered set and statements of set theory is introduced in this chapter and used to calculate truth values in V(B). This leads to the construction ... More


Which Undecidable Mathematical Sentences Have Determinate Truth Values?

Hartry Field

in Truth and the Absence of Fact

Published in print:
2001
Published Online:
November 2003
ISBN:
9780199242894
eISBN:
9780191597381
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0199242895.003.0012
Subject:
Philosophy, General

Argues that typical undecidable sentences of set theory (e.g. about the size of the continuum) can have no determinate truth‐value, since nothing in our practice can determine which ‘universe of ... More


Indeterminacy, Degree of Belief, and Excluded Middle

Hartry Field

in Truth and the Absence of Fact

Published in print:
2001
Published Online:
November 2003
ISBN:
9780199242894
eISBN:
9780191597381
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0199242895.003.0010
Subject:
Philosophy, General

Discusses the question of how to reconcile the acceptance of indeterminacy in one's own language with the acceptance of a minimal notion of truth. Argues that regarding a sentence of one's own ... More


Mathematical Objectivity and Mathematical Objects

Hartry Field

in Truth and the Absence of Fact

Published in print:
2001
Published Online:
November 2003
ISBN:
9780199242894
eISBN:
9780191597381
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0199242895.003.0011
Subject:
Philosophy, General

Focuses on an issue about the objectivity of mathematics—the extent to which undecidable sentences have determinate truth‐value—and argues that this issue is more important than the issue of the ... More


Join the community, with Hilbert’s twenty-three problems

Susan D'Agostino

in How to Free Your Inner Mathematician: Notes on Mathematics and Life

Published in print:
2020
Published Online:
April 2020
ISBN:
9780198843597
eISBN:
9780191879388
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198843597.003.0021
Subject:
Mathematics, Educational Mathematics, Applied Mathematics

“Join the community, with Hilbert’s twenty-three problems” tells the story of German mathematician David Hilbert who, in 1900 at the International Congress of Mathematicians in Paris, presented ... More


Internal categoricity and the sets

Tim Button and Sean Walsh

in Philosophy and Model Theory

Published in print:
2018
Published Online:
May 2018
ISBN:
9780198790396
eISBN:
9780191863424
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198790396.003.0011
Subject:
Philosophy, Logic/Philosophy of Mathematics

As the previous chapter discussed the internalist perspective on the categoricity of arithmetic, this chapter presents the internalist perspective on sets. In particular, we show both how to ... More


Fundamentals of Fluid Dynamics

Anatoly I. Ruban and Jitesh S. B. Gajjar

in Fluid Dynamics: Part 1: Classical Fluid Dynamics

Published in print:
2014
Published Online:
August 2014
ISBN:
9780199681730
eISBN:
9780191761607
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199681730.003.0002
Subject:
Physics, Soft Matter / Biological Physics

This chapter starts with a discussion of the continuum hypothesis and the conditions under which it can be used. The origin of the internal forces acting in moving fluids is then identified and the ... More


Conclusion

Ian Rumfitt

in The Boundary Stones of Thought: An Essay in the Philosophy of Logic

Published in print:
2015
Published Online:
June 2015
ISBN:
9780198733638
eISBN:
9780191798016
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198733638.003.0010
Subject:
Philosophy, Philosophy of Mind

In defending classical logic against some of the attacks analysed in previous chapters, it was conceded that they constitute powerful objections to principles of classical semantics, notably the ... More


Properties of first-order logic

Shawn Hedman

in A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity

Published in print:
2004
Published Online:
November 2020
ISBN:
9780198529804
eISBN:
9780191916656
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198529804.003.0008
Subject:
Computer Science, Computer Architecture and Logic Design

We show that first-order logic, like propositional logic, has both completeness and compactness. We prove a countable version of these theorems in Section 4.1. We ... More


Systems of Logic Based on Ordinals (1938)

Alan Turing

in The Essential Turing

Published in print:
2004
Published Online:
November 2020
ISBN:
9780198250791
eISBN:
9780191916526
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198250791.003.0007
Subject:
Computer Science, History of Computer Science

On 23 September 1936 Turing left England on a vessel bound for New York. His destination was Princeton University, where the Mathematics Department and the ... More


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