Understanding the InfiniteHarvard University Press, 2009. gada 30. jūn. - 384 lappuses An accessible history and philosophical commentary on our notion of infinity. How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge. Praise for Understanding the Infinite “Understanding the Infinite is a remarkable blend of mathematics, modern history, philosophy, and logic, laced with refreshing doses of common sense. It is a potted history of, and a philosophical commentary on, the modern notion of infinity as formalized in axiomatic set theory . . . An amazingly readable [book] given the difficult subject matter. Most of all, it is an eminently sensible book. Anyone who wants to explore the deep issues surrounding the concept of infinity . . . will get a great deal of pleasure from it.” —Ian Stewart, New Scientist “How, in a finite world, does one obtain any knowledge about the infinite? Lavine argues that intuitions about the infinite derive from facts about the finite mathematics of indefinitely large size . . . The issues are delicate, but the writing is crisp and exciting, the arguments original. This book should interest readers whether philosophically, historically, or mathematically inclined, and large parts are within the grasp of the general reader. Highly recommended.” —D. V. Feldman, Choice |
Saturs
| 2 | |
| 7 | |
What Are Sets? | 28 |
1 Russell | |
1 The Axiom of Choice | |
Leaps of Faith | |
1 Intuition | |
1 Who Needs SelfEvidence? | |
1 Natural Models | |
Bibliography | |
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absolutely infinite actually analysis arbitrary argument assumption Axiom of Choice axiomatic axiomatic set theory Cantor Cantorian cardinal combinatorial collections construction Continuum Hypothesis convergence counted curve defined definition denumerable discussed domain example extrapolation fact Fin(T Fin(ZFC finitary mathematics finite mathematics finite set theory first-order first-order logic formalism Foundation Fourier Fraenkel Frege given Gödel Grundlagen Hilbert idea indefinitely large infinitary infinite sets infinity introduced intuition intuitionist irrational numbers iterative conception Leibniz limit logical collections mathematical objects mathematicians natural model natural numbers Neumann notion number class ordinal numbers paradoxes Peano arithmetic philosophy of mathematics Power Set predecessors primitive recursive principle problem proof propositional functions prove quantifiers rational numbers real numbers relation Replacement result Russell Russell's schematic second-order schematic variables second-order logic second-order set theory self-evident sense sentence sequence set-theoretic Skolem subset symbols theorem transfinite trigonometric series type and order upper bound values well-ordered set Zermelo