LEADER 03627nam 2200601 450 001 996466668903316 005 20220428130705.0 010 $a1-280-62578-3 010 $a9786610625789 010 $a3-540-34268-0 024 7 $a10.1007/11601562 035 $a(CKB)1000000000232744 035 $a(DE-He213)978-3-540-34268-7 035 $a(SSID)ssj0000108244 035 $a(PQKBManifestationID)11145230 035 $a(PQKBTitleCode)TC0000108244 035 $a(PQKBWorkID)10044726 035 $a(PQKB)11575194 035 $a(MiAaPQ)EBC3036438 035 $a(MiAaPQ)EBC6699020 035 $a(Au-PeEL)EBL6699020 035 $a(PPN)123130085 035 $a(EXLCZ)991000000000232744 100 $a20220428d2006 uy 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAxiom of choice /$fHorst Herrlich 205 $a1st ed. 2006. 210 1$aBerlin, Germany ;$aNew York, New York :$cSpringer,$d[2006] 210 4$d©2006 215 $a1 online resource (XIV, 194 p. Also available online.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v1876 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-540-30989-6 320 $aIncludes bibliographical references and index. 327 $aOrigins: Hilbert's First Problem -- Choice Principles: Some Equivalents to the Axiom of Choice, Some Concepts Related to the Axiom of Choice -- Elementary Observations: Hidden Choice, Unnecessary Choice, Concepts Split Up: Compactness -- Disasters without Choice: Finiteness, Disasters in Cardinal Arithmetic, Disasters in Order Theory, Disasters in Algebra I: Vector Spaces, Disasters in Algebra II: Categories, Disasters in Elementary Analysis: The Reals and Continuity, Disasters in Topology I: Countable Sums, Disasters in Topology II: Products (The Tychonoff and the Cech-Stone Theorem), Disasters in Topology III: Function Spaces (The Ascoli Theorem), Disasters in Topology IV: The Baire Category Theorem, Disasters in Graph Theory: Coloring Problems -- Disasters with Choice: Disasters in Elementary Analysis, Disasters in Geometry: Paradoxical Decompositions -- Disasters either way: Disasters in Game Theory -- Beauty without Choice: Lindelöf = Compact, Measurability (The Axiom of Determinateness). 330 $aAC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that: Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC). Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games). Some beautiful mathematical theorems hold only if AC is replaced by some alternative axiom, contradicting AC (e.g., by AD, the axiom of determinateness). Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v1876 606 $aAxiom of choice 615 0$aAxiom of choice. 676 $a511.3 700 $aHerrlich$b Horst$058859 712 02$aSpringerLink (Online service) 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996466668903316 996 $aAxiom of choice$9230568 997 $aUNISA