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Roads to infinity = the mathematics of truth and proof /

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Roads to infinity/ John Stillwell.
其他題名:	the mathematics of truth and proof /
作者:	Stillwell, John.
出版者:	Natick, Mass. :A K Peters, : ©2010.,
面頁冊數:	1 online resource (xi, 203 p.) :ill. :
標題:	Set theory. -
電子資源:	http://www.crcnetbase.com/doi/book/10.1201/b11162
ISBN:	9781439865507 (electronic bk.)

Roads to infinity = the mathematics of truth and proof /
Stillwell, John.

Roads to infinitythe mathematics of truth and proof /[electronic resource] :John Stillwell. - Natick, Mass. :A K Peters,©2010. - 1 online resource (xi, 203 p.) :ill.

Includes bibliographical references and index.

The diagonal argument : Counting and countability ; Does one infinite size fit all? ; Cantor's diagonal argument ; Transcendental numbers ; Other uncountability proofs ; Rates of growth ; The cardinality of the continuum ; Historical background -- Ordinals : Counting past infinity ; The countable ordinals ; The axiom of choice ; The continuum hypothesis ; Induction ; Cantor normal form ; Goodstein's Theorem ; Hercules and the Hydra ; Historical background -- Computability and proof : Formal systems ; Post's approach to incompleteness ; Gödel's first incompleteness theorem ; Gödel's second incompleteness theorem ; Formalization of computability ; The halting problem ; The entscheidungsproblem ; Historical background -- Logic : Propositional logic ; A classical system ; A cut-free system for propositional logic ; Happy endings ; Predicate logic ; Completeness, consistency, happy endings ; Historical background -- Arithmetic : How might we prove consistency? ; Formal arithmetic ; The systems PA and PA ; Embedding PA and PA; Cut elimination in PA ; The height of this great argument ; Roads to infinity ; Historical background -- Natural unprovable sentences : A generalized Goodstein Theorem ; Countable ordinals via natural numbers ; From generalized Goodstein to well-ordering ; Generalized and ordinary Goodstein ; Provably computable functions ; Complete disorder is impossible ; The hardest theorem in graph theory ; Historical background -- Axioms of infinity : Set theory without infinity ; Inaccessible cardinals ; The axiom of determinacy ; Largeness axioms for arithmetic ; Large cardinals and finite mathematics ; Historical background.

Offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. From publisher description.

ISBN: 9781439865507 (electronic bk.)Subjects--Topical Terms:

579942
Set theory.

LC Class. No.: QA248 / .S778 2010

Dewey Class. No.: 511.3/22

Roads to infinity = the mathematics of truth and proof /
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504 $a Includes bibliographical references and index.
505 0 $6 880-01 $a The diagonal argument : Counting and countability ; Does one infinite size fit all? ; Cantor's diagonal argument ; Transcendental numbers ; Other uncountability proofs ; Rates of growth ; The cardinality of the continuum ; Historical background -- Ordinals : Counting past infinity ; The countable ordinals ; The axiom of choice ; The continuum hypothesis ; Induction ; Cantor normal form ; Goodstein's Theorem ; Hercules and the Hydra ; Historical background -- Computability and proof : Formal systems ; Post's approach to incompleteness ; Gödel's first incompleteness theorem ; Gödel's second incompleteness theorem ; Formalization of computability ; The halting problem ; The entscheidungsproblem ; Historical background -- Logic : Propositional logic ; A classical system ; A cut-free system for propositional logic ; Happy endings ; Predicate logic ; Completeness, consistency, happy endings ; Historical background -- Arithmetic : How might we prove consistency? ; Formal arithmetic ; The systems PA and PA ; Embedding PA and PA; Cut elimination in PA ; The height of this great argument ; Roads to infinity ; Historical background -- Natural unprovable sentences : A generalized Goodstein Theorem ; Countable ordinals via natural numbers ; From generalized Goodstein to well-ordering ; Generalized and ordinary Goodstein ; Provably computable functions ; Complete disorder is impossible ; The hardest theorem in graph theory ; Historical background -- Axioms of infinity : Set theory without infinity ; Inaccessible cardinals ; The axiom of determinacy ; Largeness axioms for arithmetic ; Large cardinals and finite mathematics ; Historical background.
520 $a Offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. From publisher description.
588 0 $a Print version record.
650 0 $a Set theory. $3 579942
650 0 $a Infinite. $3 997224
650 0 $a Logic, Symbolic and mathematical. $3 527823
856 4 0 $u http://www.crcnetbase.com/doi/book/10.1201/b11162