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Kalenderblätter 1001 - 1111 Kalenderblätter 1001 - 1111 Ich übergebe sie mit zweifelhaften Gefühlen der Öffentlichkeit. Daß es dieser Arbeit in ihrer Dürftigkeit und der Finsternis dieser Zeit beschieden sein sollte, Licht in ein oder das andere Gehirn zu werfen, ist nicht unmöglich; aber freilich nicht wahrscheinlich. [Ludwig Wittgenstein: Philosophische Untersuchungen] 1001 120301 Mathematik für die ersten Semester: Top 10 1002 120302 The devil's offer 1003 120303 Enderton: Formulierungen des Auswahlaxioms 1004 120304 Heuser, Antisthenes 1005 120305 Rucker 1006 120306 Cantors Weltbild (33): Fest und unerschütterlich 1007 120307 V und Ù 1008 120308 Konstruierbar und unkonstruierbar 1009 120309 Moore 1010 120310 Cantors Weltbild (34): Verallgemeinerung des Zahlbegriffs 1011 120311 Syllogismus auf Griechisch 1012 120312 Grattan 1013 120313 Dummett 1014 120314 7. Auflage der Geschichte des Unendlichen, 3 Exemplare zu verschenken 1015 120315 Engel und große Kardinalzahlen 1016 120316 Opinions on continuum hypothesis and inaccessible cardinals 1017 120317 Schröter über Fischer 1018 120318 Fischer 1019 120319 Fischer 1020 120320 Fischer 1021 120321 Hilbert, Slater 1022 120322 Voltaire 1023 120323 Tait 1024 120324 Moore 1025 120325 Nietzsche 1026 120326 Taschner 1027 120327 Boolos 1028 120328 Kant, Galletti 1029 120329 Cantors Weltbild (35): Racenantisemitismus 1030 120330 Nietzsche über das Christentum, Cantor 1031 120331 Ramsey über Formalismus 1032 120401 In Sachen Georg Cantor 1033 120402 Corazza 1034 120403 Cattabriga 1035 120404 Mengenfolgen und Supertasks (1) 1036 120405 Mengenfolgen und Supertasks (2) 1037 120406 Mengenfolgen und Supertasks (3) 1038 120407 Mengenfolgen und Supertasks (4) 1039 120408 Mengenfolgen und Supertasks (5) 1040 120409 Mengenfolgen und Supertasks (6) 1041 120410 Mengenfolgen und Supertasks (7) 1042 120411 Mengenfolgen und Supertasks (8) 1043 120412 Mengenfolgen und Supertasks (9) 1044 120413 Mengenfolgen und Supertasks (10) 1045 120414 Kant, Gladis, Rosenthal 1046 120415 Gödels Geist in Ruckers Mindscape 1047 120416 Problem mit dem Begriff der Abzählbarkeit 1048 120417 Pascal 1049 120418 Slater, Koskensilta 1050 120419 Ramsey, Priest 1051 120420 Laotse, Adams, Schinkensemmel-Assoziation 1052 120421 Cantors Weltbild (36): Vom Sein des Seins - Heidegger, Sartre 1053 120422 Anwendungen der Mengenlehre (1) 1054 120423 Anwendungen der Mengenlehre (2) 1055 120424 Anwendungen der Mengenlehre (3) 1056 120425 Anwendungen der Mengenlehre (4) 1057 120426 Anwendungen der Mengenlehre (5) 1058 120427 Anwendungen der Mengenlehre (6) 1059 120428 Anwendungen der Mengenlehre (7) 1060 120429 Anwendungen der Mengenlehre (8) 1061 120430 Hume 1062 120501 Descartes 1063 120502 Spinoza, Gottesbeweis aus der Mengenlehre 1064 120503 Locke, über ω hinweg 1065 120504 Potentiell versus aktual (1) 1066 120505 Potentiell versus aktual (2) 1067 120506 Potentiell versus aktual (3) 1068 120507 Potentiell versus aktual (4) 1069 120508 Potentiell versus aktual (5) 1070 120509 Potentiell versus aktual (6) 1071 120510 Potentiell versus aktual (7) 1072 120511 Potentiell versus aktual (8) 1073 120512 Potentiell versus aktual (9) 1074 120513 Potentiell versus aktual (10) 1075 120514 Potentiell versus aktual (11) 1076 120515 Cantors Weltbild (37): Die Realität der Mathematik 1077 120516 Cantors Weltbild (38): Die Realität transfiniter Zahlen 1078 120517 Novalis 1079 120518 Undefinierbare Definitionen können geordnet werden 1080 120519 Rucker: Gödel 1081 120520 Hegel 1082 120521 Hegel 1083 120522 Hilbert, Slater 1084 120523 Cantor: Axiome 1085 120524 Reeken, Robinson, Gödel 1086 120525 Cantors Weltbild (39): Giganten 1087 120526 Abian: Inaccessible cardinal numbers, Widerspruch in ZFC 1088 120527 Cantors Pfingstepistel 1089 120528 St. Thomas Aquinas: Von der Menge der Engel 1090 120529 Die Hierarchien der Engel und der Kardinalzahlen 1091 120530 Anti-Überabzählbarkeitsbeweis 1092 120531 Grenzwert geschlossener Intervalle, Grelling-Nelson Antinomie 1093 120601 Rückblick 1094 120602 Locke, Mill 1095 120603 Slater, Priest 1096 120604 Rucker 1097 120605 Sollten Zahlen definierbar sein? 1098 120606 Cantors Weltbild (40): Anglophilie 1099 120607 Are real numbers countable in constructive mathematics? 1100 120608 Ramsey Priest 1101 120609 Moore 1102 120610 Open letter 1103 120611 MatheRealismus (1) 1104 120612 MatheRealismus (2) 1105 120613 MatheRealismus (3) 1106 120614 MatheRealismus (4) 1107 120615 MatheRealismus (5) 1108 120616 MatheRealismus (6) 1109 120617 MatheRealismus (7) 1110 120618 MatheRealismus (8) 1111 120619 Valet! 1001 Das Kalenderblatt 120301 "Mathematik für die ersten Semester" http://www.hs-augsburg.de/einrichtung/presse/mitteilungen_2012/feb/2012_02_03/index.html eines der erfolgreichsten Mathematiklehrbücher http://issuu.com/oldenbourg/docs/owv_mint_verlagsvorschau_120109_web/42?mode=a_p ist in dritter Auflage erschienen: http://www.oldenbourg-verlag.de/wissenschaftsverlag/mathematik-ersten- semester/9783486708219 1002 Das Kalenderblatt 120302 Ms C dies and goes to hell, or to a place that seems like hell. The devil approaches and offers to play a game of chance. If she wins, she can go to heaven. If she loses, she will stay in hell forever; there is no second chance to play the game. If Ms C plays today, she has a 1/2 chance of winning. Tomorrow the probability will be 2/3. Then 3/4, 4/5, 5/6, etc., with no end to the series. Thus every passing day increases her chances of winning. At what point should she play the game? The answer is not obvious: after any given number of days spent waiting, it will still be possible to improve her chances by waiting yet another day. And any increase in the probability of winning a game with infinite stakes has an infinite utility. For example, if she waits a year, her probability of winning the game would be approximately .997268; if she waits one more day, the probability would increase to .997275, a difference of only .000007. Yet, even .000007 multiplied by infinity is infinite. On the other hand, it seems reasonable to suppose the cost of delaying for a day to be finite - a day's more suffering in hell. So the infinite expected benefit from a delay will always exceed the cost. This logic might suggest that Ms C should wait forever, but clearly such a strategy would be self defeating: why should she stay forever in a place in order to increase her chances of leaving it? So the question remains: what should Ms C do?' [E.J. Gracely: "Playing games with eternity: The devil's offer", Analysis 48.3 (1988) p. 113] http://www.balliol.ox.ac.uk/sites/default/files/Dudman-1988-Indicative-and-Subjunctive.pdf Die Sentenz zu ewiger Verdammnis - eine der wenigen praktischen Anwendungen der transfiniten Mengenlehre. 1003 Das Kalenderblatt 120303 The following statements are equivalent. (1) Axiom of choice, I. For any relation R there is a function F Œ R with domF = domR. (2) Axiom of choice, II; multiplicative axiom: The Cartesian product of nonempty sets is always nonempty. That is, if H is a function with domain I and if (" i œ I) H(i) ∫ «, then there is a function f with domain I such that (" i œ I) f(i) œ H(i). (3) Axiom of choice, III. For any set A there is a function F (a "choice function" for A) such that the domain of F is the set of nonempty subsets of A, and such that F(B) œ B for every B Œ A. (4) Axiom of choice, IV. Let A be a set such that (a) each member of A is a nonempty set, and (b) any two distinct members of A are disjoint. Then there exists a set C containing exactly one element from each member of A (i.e., for each B œ A the set C … B is a singleton {x} for some x). (5) Cardinal comparability. For any sets C and D, either C Ç D or D í C. For any two cardinal numbers κ and λ, either κ § λ or k ¥ λ. (6) Zorn's lemma. Let A be a set such that for every chain B Œ A, we have »B œ A. (B is called a chain iff for any C and D in B, either C Œ D or D Œ C.) Then A contains an element M (a "maximal" element) such that M is not a subset of any other set in A. (6) fl (1) The strategy behind this application (and others) of Zorn's lemma is to form a collection A of pieces of the desired object, and then to show that a maximal piece serves the intended purpose. In the present case, we are given a relation R and we choose to define A = {f Œ R | f is a function}. Before we can appeal to Zorn's lemma, we must check that A is closed under unions of chains. So consider any chain B Œ A. Since every member of B is a subset of R, »B is a subset of R. To see that »B is a function, we use the fact that B is a chain. lf ‚x, yÚ and ‚x, zÚ belong to »B, then ‚x, yÚ œ G œ B and ‚x, zÚ œ H œ B for some functions G and H in A. Either G Œ H or H Œ G; in either event both ‚x, yÚ and ‚x, zÚ belong to a single function, so y = z. Hence »B is in A. Now we can appeal to (6), which provides us wich a maximal Function F in A. We claim that domF = domR. For otherwise take any x œ domR \ domF. Since x œ domR, there is some y with xRy. Define F' = F » {‚x, yÚ}. Then F' œ A is contradicting the maximality of F. Hence domF = domR. We can give a plausibility argument for Zorn's lemma as follows. A cannot be empty, because « is a chain and so «= »« œ A. Probably « is not maximal, so we can choose a larger set. If that larger set is not maximal, we can choose a still larger one. After infinitely many steps, even if we have not found a maximal set we have at least formed a chain. So we can take its union and continue.
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