000 04152nam a22005775i 4500
001 978-3-540-34268-7
003 DE-He213
005 20190213151648.0
007 cr nn 008mamaa
008 100301s2006 gw | s |||| 0|eng d
020 _a9783540342687
_9978-3-540-34268-7
024 7 _a10.1007/11601562
_2doi
050 4 _aQA8.9-10.3
072 7 _aPBC
_2bicssc
072 7 _aMAT018000
_2bisacsh
072 7 _aPBC
_2thema
072 7 _aPBCD
_2thema
082 0 4 _a511.3
_223
100 1 _aHerrlich, Horst.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
245 1 0 _aAxiom of Choice
_h[electronic resource] /
_cby Horst Herrlich.
264 1 _aBerlin, Heidelberg :
_bSpringer Berlin Heidelberg,
_c2006.
300 _aXIV, 198 p. 1 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v1876
505 0 _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).
520 _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.
650 0 _aLogic, Symbolic and mathematical.
650 0 _aAlgebra.
650 0 _aFunctional analysis.
650 0 _aMathematics.
650 0 _aCombinatorics.
650 0 _aTopology.
650 1 4 _aMathematical Logic and Foundations.
_0http://scigraph.springernature.com/things/product-market-codes/M24005
650 2 4 _aGeneral Algebraic Systems.
_0http://scigraph.springernature.com/things/product-market-codes/M1106X
650 2 4 _aFunctional Analysis.
_0http://scigraph.springernature.com/things/product-market-codes/M12066
650 2 4 _aGame Theory, Economics, Social and Behav. Sciences.
_0http://scigraph.springernature.com/things/product-market-codes/M13011
650 2 4 _aCombinatorics.
_0http://scigraph.springernature.com/things/product-market-codes/M29010
650 2 4 _aTopology.
_0http://scigraph.springernature.com/things/product-market-codes/M28000
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783540819066
776 0 8 _iPrinted edition:
_z9783540309895
830 0 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v1876
856 4 0 _uhttps://doi.org/10.1007/11601562
912 _aZDB-2-SMA
912 _aZDB-2-LNM
999 _c11454
_d11454