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===위키데이터===
 
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* ID :  [https://www.wikidata.org/wiki/Q208416 Q208416]
 
* ID :  [https://www.wikidata.org/wiki/Q208416 Q208416]
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===Spacy 패턴 목록===
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* [{'LOWER': 'continuum'}, {'LEMMA': 'hypothesis'}]
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* [{'LOWER': 'hilbert'}, {'LOWER': "'s"}, {'LOWER': 'first'}, {'LEMMA': 'problem'}]
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* [{'LEMMA': 'CH'}]

2021년 2월 17일 (수) 01:02 기준 최신판

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  1. The continuum hypothesis appears in a memoir of Cantor (dated Halle a.[1]
  2. One of these models, the fast-slow continuum hypothesis, emphasizes the role played by mortality at different stages of the life cycle in shaping the large array of life history variation.[2]
  3. CH is simply the statement that there’s no infinity intermediate between ℵ 0 and C: that anything greater than the first is at least the second.[3]
  4. Cantor tried in vain for decades to prove or disprove CH; the quest is believed to have contributed to his mental breakdown.[3]
  5. Halfway between Hilbert’s speech and today, the question of CH was finally “answered,” with the solution earning the only Fields Medal that’s ever been awarded for work in set theory and logic.[3]
  6. The easier half, the consistency of CH with set theory, was proved by incompleteness dude Kurt Gödel in 1940; the harder half, the consistency of not(CH), by Paul Cohen in 1963.[3]
  7. Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis.[4]
  8. Most of the first part is devoted to "plausible arguments" for or against CH: how it stands relative to both other possible axioms and to various set-theoretic "rules of thumb".[4]
  9. Most of those who disbelieve CH think that the continuum is likely to have very large cardinality, rather than aleph_2 (as Gödel seems to have suggested).[4]
  10. Also, CH is much easier to force (Cohen's method) than ~CH.[4]
  11. Kurt Gödel showed in 1940 the relative consistency of ZFC along with CH.[5]
  12. That is, he built a model for set theory where CH is true.[5]
  13. On the other hand, in 1963 Paul Cohen developed a new technique called forcing which builds models of set theory where ZFC holds but CH fails.[5]
  14. theorist Hugh Woodin has set out a mathematical program to investigate CH by investigating what are called large cardinal axioms.[5]
  15. The continuum problem asks for a solution of the continuum hypothesis (CH), and is the first in Hilbert's celebrated list of 23 problems.[6]
  16. showed that the accepted axioms of set theory cannot solve CH.[6]
  17. Gödel in 1939 was able to produce a model of set theory in which CH is true.[6]
  18. These imply the axiom of projective determinacy and the failure of CH, as well as many new and surprising results in general topology and infinite combinatorics.[6]
  19. His 1940 book, better known by its short title, The Consistency of the Continuum Hypothesis, is a classic of modern mathematics.[7]
  20. The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers.[7]
  21. This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski.[8]
  22. We use Boolean-valued models to give forcing arguments for both directions, using Cohen forcing for the consistency of ¬ CH and a σ-closed forcing for the consistency of CH.[9]
  23. Most of the first part is devoted to ``plausible arguments" for or against CH: how it stands relative to both other possible axioms and to various set-theoretic ``rules of thumb".[10]
  24. Most of those who disbelieve CH think that the continuum is likely to have very large cardinality, rather than (as Godel seems to have suggested).[10]
  25. Also, CH is much easier to force (Cohen's method) than CH.[10]
  26. And CH is much more likely to settle various outstanding results than is CH, which tends to be neutral on these results.[10]
  27. D. Hilbert posed, in his celebrated list of problems, as Problem 1 that of proving Cantor's continuum hypothesis (the problem of the continuum).[11]
  28. he had proved using the Continuum Hypothesis in order to answer negatively a question of Erdoos and Fodor.[12]
  29. The continuum hypothesis is a famous problem of set theory concerning the cardinality of the real numbers (the “continuum”).[13]
  30. accordingly, viewed Gödel’s 1938 result as a proof of CH CH , whereas in P. Dehornoy‘s 2003 reinterpretation based on work of Woodin, CH CH is actually conjectured to be false.[13]
  31. Regarding the statement of the generalized continuum hypothesis in ZF (not ZFC), one should distinguish various possibilities.[13]
  32. But then one could argue such a generalized continuum hypothesis is not as general or strong as it might be, since not all sets can be well-ordered using ZF alone.[13]
  33. Continuum hypothesis, statement of set theory that the set of real numbers (the continuum) is in a sense as small as it can be.[14]
  34. The problem known as the continuum hypothesis has had perhaps the strangest fate of all.[15]
  35. How ironic then that the continuum hypothesis is unsolvable—indeed, “provably unsolvable,” as we say.[15]
  36. This means that none of the known mathematical methods—those that mathematicians actually use and find legitimate—will suffice to settle the continuum hypothesis one way or another.[15]
  37. I will explain some of these developments, along with some of the more recent history of the continuum hypothesis, from the point of view of Kurt Gödel’s role in them.[15]
  38. In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.[16]
  39. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900.[16]
  40. Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it (Dauben 1990).[16]
  41. Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory.[16]
  42. The continuum hypothesis asserts that there is no infinity between the smallest kind — the set of counting numbers — and what it asserts is the second-smallest — the continuum.[17]
  43. “If forcing axioms are right, then the continuum hypothesis is false,” Koellner said.[17]
  44. Meanwhile, forcing axioms, which deem the continuum hypothesis false by adding a new size of infinity, would also extend the frontiers of mathematics in other directions.[17]
  45. According to some theorists, there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false — but all equally worth exploring.[17]
  46. Gödel showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel set theory.[18]
  47. However, using a technique called forcing, Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory.[18]
  48. Conway and Guy (1996, p. 282) recount a generalized version of the continuum hypothesis originally due to Hausdorff in 1908 which is also undecidable: is for every ?[18]
  49. Woodin (2001ab, 2002) formulated a new plausible "axiom" whose adoption (in addition to the Zermelo-Fraenkel axioms and axiom of choice) would imply that the continuum hypothesis is false.[18]
  50. Here is an overview of the entry: Section 1 surveys the independence results in cardinal arithmetic, covering both the case of regular cardinals (where CH lies) and singular cardinals.[19]
  51. Section 2 considers approaches to CH where one successively verifies a hierarchy of approximations to CH, each of which is an “effective” version of CH.[19]
  52. The centerpiece of the discussion is the discovery of a “canonical” model in which CH fails.[19]
  53. This formed the basis of a network of results that was collectively presented by Woodin as a case for the failure of CH.[19]

소스

  1. ↑ Continuum hypothesis
  2. ↑ Life history variation in plants: an exploration of the fast-slow continuum hypothesis
  3. ↑ 3.0 3.1 3.2 3.3 Shtetl-Optimized » Blog Archive » The Complete Idiot’s Guide to the Independence of the Continuum Hypothesis: Part 1 of
  4. ↑ 4.0 4.1 4.2 4.3 Continuum Hypothesis: True, False, or Neither?
  5. ↑ 5.0 5.1 5.2 5.3 The Intrepid Mathematician
  6. ↑ 6.0 6.1 6.2 6.3 The Continuum Hypothesis: A Mystery of Mathematics?
  7. ↑ 7.0 7.1 Consistency of the Continuum Hypothesis. (AM-3), Volume 3
  8. ↑ Chance and the Continuum Hypothesis
  9. ↑ A formal proof of the independence of the continuum hypothesis
  10. ↑ 10.0 10.1 10.2 10.3 The Continuum Hypothesis
  11. ↑ Encyclopedia of Mathematics
  12. ↑ Continuum Hypothesis - an overview
  13. ↑ 13.0 13.1 13.2 13.3 continuum hypothesis in nLab
  14. ↑ Continuum hypothesis | mathematics
  15. ↑ 15.0 15.1 15.2 15.3 Can the Continuum Hypothesis Be Solved?
  16. ↑ 16.0 16.1 16.2 16.3 Continuum hypothesis
  17. ↑ 17.0 17.1 17.2 17.3 Quanta Magazine
  18. ↑ 18.0 18.1 18.2 18.3 Continuum Hypothesis -- from Wolfram MathWorld
  19. ↑ 19.0 19.1 19.2 19.3 The Continuum Hypothesis (Stanford Encyclopedia of Philosophy)

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Spacy 패턴 목록

  • [{'LOWER': 'continuum'}, {'LEMMA': 'hypothesis'}]
  • [{'LOWER': 'hilbert'}, {'LOWER': "'s"}, {'LOWER': 'first'}, {'LEMMA': 'problem'}]
  • [{'LEMMA': 'CH'}]
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