## 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

## Mathematical Justifications are Not Infinitely Various

*Neil Tennant*

### in Changes of Mind: An Essay on Rational Belief Revision

- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199655755
- eISBN:
- 9780191742125
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199655755.003.0010
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

This chapter gives a suitably texturized proof of a deep result in mathematical logic by Harvey Friedman, which was produced on request. It states that every extant mathematical theory (by virtue of ... More

## It stands alone

*Louis A. Girifalco*

### in The Universal Force: Gravity - Creator of Worlds

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228966
- eISBN:
- 9780191711183
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228966.003.0016
- Subject:
- Physics, History of Physics

The theory of relativity is unique and is based on very simple ideas. In fact, it has only one fundamental premise and one physical fact. The premise is that the laws of physics must be the same ... More

## Set Theory and its Philosophy: A Critical Introduction

*Michael Potter*

- Published in print:
- 2004
- Published Online:
- September 2011
- ISBN:
- 9780199269730
- eISBN:
- 9780191699443
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269730.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

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 ... 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

## Mathematics without Numbers: Towards a Modal-Structural Interpretation

*Geoffrey Hellman*

- Published in print:
- 1993
- Published Online:
- November 2003
- ISBN:
- 9780198240341
- eISBN:
- 9780191597664
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198240341.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Develops a structuralist understanding of mathematics, as an alternative to set‐ or type‐theoretic foundations, that respects classical mathematical truth while minimizing Platonist commitments to ... More

## Conclusions: A New Basis for Interpreting the Mysteries

*ROGER BECK*

### in The Religion of the Mithras Cult in the Roman Empire: Mysteries of the Unconquered Sun

- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199216130
- eISBN:
- 9780191712128
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199216130.003.0012
- Subject:
- Classical Studies, Ancient Religions

This concluding chapter revisits the summary description of the religion of the Mithras cult proposed in Chapter 1.

## Among

*Thomas J. McKay*

### in Plural Predication

- Published in print:
- 2006
- Published Online:
- January 2007
- ISBN:
- 9780199278145
- eISBN:
- 9780191707971
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199278145.003.0007
- Subject:
- Philosophy, Philosophy of Language

This chapter develops some axioms that express the fundamental features of the among relationship. It shows that although among is interdefinable with one of on one understanding of the one of ... More

## Problems

*David Bostock*

### in Russell's Logical Atomism

- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199651443
- eISBN:
- 9780191741197
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199651443.003.0006
- Subject:
- Philosophy, History of Philosophy, Metaphysics/Epistemology

Russell’s attempt to obey the Vicious Circle Principle must lead to the conclusion that there are only countably many propositional functions, but this conflicts with his axiom of reducibility. So we ... More

## The Problem of Conceptual Understanding

*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/9780691167510.003.0010
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter considers one of the most intriguing questions that philosophy of mathematics in practice must, sooner or later, confront: how understanding of mathematics is obtained. In particular, it ... More

## Defending the Axioms: On the Philosophical Foundations of Set Theory

*Penelope Maddy*

- Published in print:
- 2011
- Published Online:
- May 2011
- ISBN:
- 9780199596188
- eISBN:
- 9780191725395
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199596188.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

Mathematics depends on proofs, and proofs have to begin somewhere, from some fundamental assumptions. Chapter I traces the historical rise of pure mathematics and the development of set theory, ... More

## Computing Changes in Belief

*Neil Tennant*

### in Changes of Mind: An Essay on Rational Belief Revision

- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199655755
- eISBN:
- 9780191742125
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199655755.003.0002
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

This chapter introduces the basic ideas involved in our formal modeling of belief schemes as finite dependency networks. The chapter emphasizes the epistemological motivation for using each formal ... More

## Reverse Mathematics: Proofs from the Inside Out

*John Stillwell*

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691196411
- eISBN:
- 9781400889037
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691196411.001.0001
- Subject:
- Mathematics, History of Mathematics

Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have ... 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

## What Would a Scientific Economics Look Like?

*MICHAEL JOFFE*

### in Evidence, Inference and Enquiry

- Published in print:
- 2011
- Published Online:
- January 2013
- ISBN:
- 9780197264843
- eISBN:
- 9780191754050
- Item type:
- chapter

- Publisher:
- British Academy
- DOI:
- 10.5871/bacad/9780197264843.003.0016
- Subject:
- Sociology, Methodology and Statistics

This chapter compares biology with the practices of economics to determine the extent to which mainstream economic theory can be regarded as ‘scientific’. This survey of practice in the two ... More

## Implementation of the Extension of the Truth Theory to Quantifiers

*Ernest Lepore and Kirk Ludwig*

### in Donald Davidson's Truth-Theoretic Semantics

- Published in print:
- 2007
- Published Online:
- May 2008
- ISBN:
- 9780199290932
- eISBN:
- 9780191710445
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199290932.003.0004
- Subject:
- Philosophy, Philosophy of Language

This chapter gives a small informal theory for a language fragment including quantifiers in illustration. The discussion in subsequent chapters takes place, then, in this new framework in which the ... More

## CONSTRUCTIVE SET THEORIES AND THEIR CATEGORY-THEORETIC MODELS

*Alex Simpson*

### in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566519
- eISBN:
- 9780191713927
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566519.003.0003
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter advocates a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. The ... More

## THE DUALITY OF CLASSICAL AND CONSTRUCTIVE NOTIONS AND PROOFS

*Sara Negri and Jan Von Plato*

### in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566519
- eISBN:
- 9780191713927
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566519.003.0009
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

The method of converting mathematical axioms into rules of sequent calculus reveals a perfect duality between classical and constructive basic notions, such as equality and apartness, and between the ... More

## ITERATED BOOLEAN EXTENSIONS, MARTIN'S AXIOM, AND SOUSLIN'S HYPOTHESIS

*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/9780198568520.003.0007
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

The technique of iterating Boolean extensions is introduced in this chapter and applied to prove the independence of Souslin’s hypothesis concerning the structure of densely ordered sets.

## The Tractatus

*Michael Potter*

### in Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap

- Published in print:
- 2002
- Published Online:
- May 2007
- ISBN:
- 9780199252619
- eISBN:
- 9780191712647
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199252619.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Ludwig Wittgenstein studied with Russell in Cambridge from 1911 to 1913, and wrote the Tractatus while on active service in the Austrian army during the First World War. Large parts of the book are ... More