*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 ...
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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 exposition of the theory of Boolean-valued models as developed by Scott and Solovay in the 1960s, deriving along the way the central set theoretic independence proofs of Cohen and others in the particularly elegant form that the Boolean-valued approach enables them to assume. In this edition, the background material has been augmented to include an introduction to Heyting algebras. It includes chapters on Boolean-valued analysis and Heyting-algebra-valued models of intuitionistic set theory.Less

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 exposition of the theory of Boolean-valued models as developed by Scott and Solovay in the 1960s, deriving along the way the central set theoretic independence proofs of Cohen and others in the particularly elegant form that the Boolean-valued approach enables them to assume. In this edition, the background material has been augmented to include an introduction to Heyting algebras. It includes chapters on Boolean-valued analysis and Heyting-algebra-valued models of intuitionistic set theory.

*John L. Bell*

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568520
- eISBN:
- 9780191717581
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568520.003.0006
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of ...
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This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of Boolean-valued models in proving theorems about complete Boolean algebras.Less

This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of Boolean-valued models in proving theorems about complete Boolean algebras.

*Sergei Zuyev*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0016
- Subject:
- Mathematics, Geometry / Topology

Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and ...
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Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and analysis of modern telecommunications systems, in which spatial arrangement is typically a crucial consideration in their performance evaluation, optimization or future development. In this survey we aim to summarize the main stochastic geometry models and tools currently used in studying modern telecommunications. We outline specifics of wired, wireless fixed and ad hoc systems and show how stochastic geometry modelling helps in their analysis and optimization. Point and line processes, Palm theory, shot‐noise processes, random tessellations, Boolean models, percolation, random graphs and networks, spatial statistics and optimization: this is a far from exhaustive list of techniques used in studying contemporary telecommunications systems and which we shall briefly discuss.Less

Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and analysis of modern telecommunications systems, in which spatial arrangement is typically a crucial consideration in their performance evaluation, optimization or future development. In this survey we aim to summarize the main stochastic geometry models and tools currently used in studying modern telecommunications. We outline specifics of wired, wireless fixed and ad hoc systems and show how stochastic geometry modelling helps in their analysis and optimization. Point and line processes, Palm theory, shot‐noise processes, random tessellations, Boolean models, percolation, random graphs and networks, spatial statistics and optimization: this is a far from exhaustive list of techniques used in studying contemporary telecommunications systems and which we shall briefly discuss.

*Tim Button and Sean Walsh*

- Published in print:
- 2018
- Published Online:
- May 2018
- ISBN:
- 9780198790396
- eISBN:
- 9780191863424
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198790396.003.0013
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin ...
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Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.Less

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.