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  1. This shows that the assumptions on which the axiom of choice rests cannot be fully implemented in FAST.[1]
  2. Did the cardinality of the set of choices affect the validity of the Axiom, so that the Denumerable Axiom was true but not the Axiom of Choice in general?[2]
  3. The axiom of choice is extremely useful, and it seems extremely natural as well.[3]
  4. This is truly mind boggling, and a lot of people object to the axiom of choice on the ground that this process shouldn't be possible.[3]
  5. And our first important application is the introduction of a model of Zermelo-Fraenkel set theory where the Partition Principle (PP) holds but not the Axiom of Choice (AC).[4]
  6. analysis of the axiom of choice with J. Hintikka’s standing on this axiom.[5]
  7. Hintikka preferred version of the axiom of choice is indeed acceptable for the constructivists and its meaning does not involve higher order logic.[5]
  8. In all of these cases, the "axiom of choice" fails.[6]
  9. In the comments, there was a request for a totally explicit example, where Axiom of Choice is commonly used but not necessary.[6]
  10. It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis.[7]
  11. Hint: the solution uses the axiom of choice.[7]
  12. Each one states that a certain proposition im- plies the Axiom of Choice.[8]
  13. In fact, Friedrich Hartogs proved in 1915 that Trichotomy implies the Axiom of Choice.[8]
  14. The second surprise, published by Waclaw Sierpin'ski in 1947, is that the General Continuum Hypothesis implies the Axiom of Choice, whereas the two seem to have nothing to do with one another.[8]
  15. Kurt Godel proved in 1938 that the General Continuum Hypothesis and the Axiom of Choice are consistent with the usual (Zermelo-Fraenkel) axioms of set theory (4).[8]
  16. The Axiom of Choice is equivalent to Zorn's Lemma discussed at length in The Mathematics of Logic and also to Cantor's Well-Ordering principle.[9]
  17. In fact Theorem 2.36 is a thinly disguised proof taking us from the Axiom of Choice to the Zorn's Lemma via the Well-Ordering Principle.[9]
  18. The Axiom of Choice (AC) is the remaining axiom to be added to the set of Zermelo-Fraenkel axioms (ZF) making it the full theory ZFC.[9]
  19. Zermelo introduced the Axiom of Choice as an intuitively correct axiom that proved Cantor's well-ordering principle.[9]
  20. The Banach–Tarski Paradox is so contrary to our intuition that it must have some implications for the foundations of mathematics and the unrestricted use of the Axiom of Choice (AC).[10]
  21. One, known as the axiom of choice, was the same as our intuitive assumption about the dresser drawer problem.[11]
  22. But these axioms come in two distinct flavors: one with the axiom of choice and one without.[11]
  23. Both lead to perfectly good foundations for math, though the additional power of the axiom of choice allows us to go farther and do things more easily.[11]
  24. This chapter presents examples of fundamental theorems of abstract algebra and topology whose proofs use the axiom of choice.[12]
  25. Some objections to the axiom of choice are based on the fact that the axiom has paradoxical consequences.[12]
  26. The chapter also sketches the proof of this paradox to show how the axiom of choice is used and that there is nothing paradoxical about this theorem.[12]
  27. The axiom of choice was explicitly formulated by E. Zermelo (1904) and was objected to by many mathematicians.[13]
  28. Many postulates equivalent to the axiom of choice were subsequently discovered.[13]
  29. The axiom of choice does not contradict the other axioms of set theory (e.g. the system ZF) and cannot be logically deduced from them if they are non-contradictory.[13]
  30. The axiom of choice is extensively employed in classical mathematics.[13]
  31. We examine various of these weaker forms of the Axiom of Choice and study how they are related to each other.[14]
  32. The axiom of choice has many mathematically equivalent formulations, some of which were not immediately realized to be equivalent.[15]
  33. The axiom of choice has the feature—not shared by other axioms of set theory—that it asserts the existence of a set without ever specifying its elements or any definite way to select them.[15]
  34. The axiom of choice merely asserts that it has at least one, without saying how to construct it.[15]
  35. The axiom of choice is not needed for finite sets since the process of choosing elements must come to an end eventually.[15]
  36. The Axiom of Choice (AC) was formulated about a century ago, and it was controversial for a few of decades after that; it might be considered the last great controversy of mathematics.[16]
  37. When we accept or reject the Axiom of Choice, we are specifying something about which mental universe we're choosing to work in.[16]
  38. For another indication of the controversy that initially surrounded the Axiom of Choice, consider this anecdote (recounted by Jan Mycielski in Notices of the AMS vol.[16]
  39. The full strength of the Axiom of Choice does not seem to be needed for applied mathematics.[16]
  40. The Axiom of Choice (AC) is one of the most discussed axioms of mathematics, perhaps second only to Euclid's parallel postulate.[17]
  41. So: Should you accept the Axiom of Choice?[17]
  42. The upshot of this work was that there was no way to resolve the question of the truth or falsity of the axiom of choice using the axioms of ZF.[18]
  43. Nevertheless, there are mathematicians who either do not "believe in" the axiom of choice or who are interested in the logical repercussions of disallowing this axiom in their set theory.[18]
  44. It is standard for mathematicians to keep track of which results require the axiom of choice and which ones can be proved without it.[18]
  45. The rest of the axioms of ZF set theory are essentially universally accepted, so it is helpful to be precise about exactly when the axiom of choice is unavoidable.[18]
  46. The axiom of choice allows us to arbitrarily select a single element from each set, forming a corresponding family of elements (x i ) also indexed over the real numbers, with x i drawn from S i .[19]
  47. In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty.[19]
  48. and it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC).[19]
  49. One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.[19]
  50. When the full axiom of choice fails, it may still be valid for some restricted class of objects A A and/or B B .[20]
  51. In this form, the axiom of choice may look less mysterious than in its original formulation.[20]
  52. More generally still, if C C is a site, then the axiom of choice for C C may be taken to say that any cover U → X U\to X admits a section.[20]
  53. The following statements are all equivalent to the axiom of choice in Set Set (although sometimes the proof in one direction requires excluded middle).[20]
  54. An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice.[21]
  55. In 1940, Gödel proved that the axiom of choice is consistent with the axioms of von Neumann-Bernays-Gödel set theory (a conservative extension of Zermelo-Fraenkel set theory).[21]
  56. However, in 1963, Cohen (1963) unexpectedly demonstrated that the axiom of choice is also independent of Zermelo-Fraenkel set theory (Mendelson 1997; Boyer and Merzbacher 1991, pp. 610-611).[21]
  57. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident.[22]
  58. In 1904 Ernst Zermelo formulated the Axiom of Choice (abbreviated as AC throughout this article) in terms of what he called coverings (Zermelo 1904).[22]
  59. The Axiom of Choice is closely allied to a group of mathematical propositions collectively known as maximal principles.[22]
  60. Zermelo’s original form of the Axiom of Choice, AC1, can be expressed as a scheme of sentences within a suitably strengthened version of \(L\).[22]

소스

  1. ↑ Is the Axiom of Choice a Logical or Set‐Theoretical Principle?
  2. ↑ Zermelo’s Axiom of Choice
  3. ↑ 3.0 3.1 Can you explain the “Axiom of choice” in simple terms?
  4. ↑ Flow: the Axiom of Choice is independent from the Partition Principle
  5. ↑ 5.0 5.1 Hintikka’s Take on the Axiom of Choice and the Constructivist Challenge
  6. ↑ 6.0 6.1 Why worry about the axiom of choice?
  7. ↑ 7.0 7.1 Most 'unintuitive' application of the Axiom of Choice?
  8. ↑ 8.0 8.1 8.2 8.3 (PDF) SIGNIFICANCE OF THE AXIOM OF CHOICE IN MATHEMATICS
  9. ↑ 9.0 9.1 9.2 9.3 The Axiom of Choice
  10. ↑ The Role of the Axiom of Choice (Chapter 15)
  11. ↑ 11.0 11.1 11.2 No. 3101: The Axiom of Choice
  12. ↑ 12.0 12.1 12.2 About the Axiom of Choice
  13. ↑ 13.0 13.1 13.2 13.3 Axiom of choice
  14. ↑ Weak Forms of the Axiom of Choice and Partitions of Infinite Sets
  15. ↑ 15.0 15.1 15.2 15.3 Axiom of choice | set theory
  16. ↑ 16.0 16.1 16.2 16.3 Axiom of Choice
  17. ↑ 17.0 17.1 The Axiom of Choice
  18. ↑ 18.0 18.1 18.2 18.3 Axiom Of Choice
  19. ↑ 19.0 19.1 19.2 19.3 Axiom of choice
  20. ↑ 20.0 20.1 20.2 20.3 axiom of choice in nLab
  21. ↑ 21.0 21.1 21.2 Axiom of Choice -- from Wolfram MathWorld
  22. ↑ 22.0 22.1 22.2 22.3 The Axiom of Choice (Stanford Encyclopedia of Philosophy)

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