Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- Acknowledgments
- Introduction
- I Perspectives on Infinity from History
- II Perspectives on Infinity from Mathematics
- III Technical Perspectives on Infinity from Advanced Mathematics
- 4 The Realm of the Infinite
- 5 A Potential Subtlety Concerning the Distinction between Determinism and Nondeterminism
- 6 Concept Calculus: Much Better Than
- IV Perspectives on Infinity from Physics and Cosmology
- V Perspectives on Infinity from Philosophy and Theology
- Index
- References
4 - The Realm of the Infinite
Published online by Cambridge University Press: 07 June 2011
- Frontmatter
- Contents
- Contributors
- Preface
- Acknowledgments
- Introduction
- I Perspectives on Infinity from History
- II Perspectives on Infinity from Mathematics
- III Technical Perspectives on Infinity from Advanced Mathematics
- 4 The Realm of the Infinite
- 5 A Potential Subtlety Concerning the Distinction between Determinism and Nondeterminism
- 6 Concept Calculus: Much Better Than
- IV Perspectives on Infinity from Physics and Cosmology
- V Perspectives on Infinity from Philosophy and Theology
- Index
- References
Summary
Introduction
The twentieth century witnessed the development and refinement of the mathematical notion of infinity. Here, of course, I am referring primarily to the development of set theory, which is that area of modern mathematics devoted to the study of infinity. This development raises an obvious question: is there a nonphysical realm of infinity?
As is customary in modern set theory, V denotes the universe of sets. The purpose of this notation is to facilitate the (mathematical) discussion of set theory – it does not presuppose any meaning to the concept of the universe of sets.
The basic properties of V are specified by the ZFC axioms. These axioms allow one to infer the existence of a rich collection of sets, a collection that is complex enough to support all of modern mathematics (and this, according to some, is the only point of the conception of the universe of sets).
I shall assume familiarity with elementary aspects of set theory. The ordinals calibrate V through the definition of the cumulative hierarchy of sets (Zermelo 1930). The relevant definition is given as follows:
Definition 1. Define for each ordinal α a set Vα by induction on α.
(1) V0 = ∅.
(2) Vα + 1 = ℙ(Vα = { X ∣ X ⊆ Vα}.
(3) If β is a limit ordinal, then Vα = ∪{Vβ ∣ β < α}.
There is a much more specific version of the question raised concerning the existence of a nonphysical realm of infinity: is the universe of sets a nonphysical realm?
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- Information
- InfinityNew Research Frontiers, pp. 89 - 118Publisher: Cambridge University PressPrint publication year: 2011
References
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