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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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