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Examples of the the word, axiom , in a Sentence Context

The word ( axiom ), is the 12346 most frequently used in English word vocabulary

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  1. Exactly one element in common with each of the sets in X. Another equivalent, axiom ,only considers collections X that are essentially power sets of other sets:: For
  2. Though this type of deduction is less common than the type which requires the, axiom ,of choice to be true. It is possible to prove many theorems using neither the
  3. Not every situation requires the axiom of choice. For finite sets X,the, axiom ,of choice follows from the other axiom s of set theory. In that case it is
  4. From each bin. In many cases such a selection can be made without invoking the, axiom ,of choice; this is in particular the case if the number of bins is finite, or
  5. Volume as the original. The pieces in this decomposition, constructed using the, axiom ,of choice, are non-measurable sets. The majority of mathematicians accept the
  6. Of important mathematical results, such as Tychonoff's theorem, require the, axiom ,of choice for their proofs. Contemporary set theorists also study axiom s that
  7. AFC (ZF plus AC) is logically equivalent (with just the ZF axiom s) to the, axiom ,of choice, and mathematicians look for results that require the axiom of choice
  8. Of choice for its existence; every set can be well-ordered if and only if the, axiom ,of choice holds. Nonconstructive aspects A proof requiring the axiom of choice
  9. That particular case is a theorem of Carmelo–Frankel set theory without the, axiom ,of choice (ZF); it is easily proved by mathematical induction. In the even
  10. Of the similarities in the language. Regional dialects In mathematics,the, axiom ,of choice, or AC, is an axiom of set theory stating that for every family (
  11. Features),such a selection can be obtained only by invoking the, axiom ,of choice. The axiom of choice was formulated in 1904 by Ernst Carmelo.
  12. Sets, there exists a choice function f defined on X. Thus the negation of the, axiom ,of choice states that there exists a set of nonempty sets which has no choice
  13. That f (B) does not lie in B. Restriction to finite sets The statement of the, axiom ,of choice does not specify whether the collection of nonempty sets is finite or
  14. Choice says that every nonempty set has an element; this holds trivially. The, axiom ,of choice can be seen as asserting the generalization of this property, already
  15. Language. Regional dialects In mathematics, the axiom of choice, or AC, is an, axiom ,of set theory stating that for every family (S_i)_ of nonempty sets there
  16. Manner, anything that is proven to exist. This foundation rejects the full, axiom ,of choice because it asserts the existence of an object without uniquely
  17. A restricted form of it, in constructive set theory from the assumption of the, axiom ,of choice. Another argument against the axiom of choice is that it implies the
  18. Real numbers that are not Lebesgue measurable can be proven to exist using the, axiom ,of choice, but it is consistent that no such set is definable. The axiom of
  19. Also study axiom s that are not compatible with the axiom of choice, such as the, axiom ,of determinant. Unlike the axiom of choice, these alternatives are not
  20. Sense for sets of sets. With this alternate notion of choice function,the, axiom ,of choice can be compactly stated as: Every set has a choice function. Which is
  21. Carmelo–Frankel set theory (ZF),regardless of the truth or falsity of the, axiom ,of choice in that particular model. The restriction to ZF renders any claim
  22. X_i)_ of elements with x_i \in S_i for every i \in I. Informally put,the, axiom ,of choice says that given any collection of bins, each containing at least one
  23. Limiting" choice function can be constructed, in general, in ZF without the, axiom ,of choice. Examples The nature of the individual nonempty sets in the
  24. Sense that, in the presence of other basic axiom s of set theory, they imply the, axiom ,of choice and are implied by it. One variation avoids the use of choice
  25. That for every set s in X, f (s) is an element of s. With this concept,the, axiom ,can be stated:: For any set X of nonempty sets, there exists a choice function
  26. Set, a choice function just corresponds to an element, so this instance of the, axiom ,of choice says that every nonempty set has an element; this holds trivially.
  27. Choice of an element from each set, and makes it unnecessary to apply the, axiom ,of choice. The difficulty appears when there is no natural choice of elements
  28. Uncountable. Hence, S breaks up into uncountable many orbits under G. Using the, axiom ,of choice, we could pick a single point from each orbit, obtaining an
  29. Theory from the assumption of the axiom of choice. Another argument against the, axiom ,of choice is that it implies the existence of counterintuitive objects. One
  30. If the axiom of choice holds. Nonconstructive aspects A proof requiring the, axiom ,of choice is nonconstructive: even though the proof establishes the existence
  31. Individual nonempty sets in the collection may make it possible to avoid the, axiom ,of choice even for certain infinite collections. For example, suppose that each
  32. Countable family of nonempty sets has a choice function, as is asserted by the, axiom ,of countable choice. If the method is applied to an infinite sequence (Xi: in
  33. Contemporary set theorists also study axiom s that are not compatible with the, axiom ,of choice, such as the axiom of determinant. Unlike the axiom of choice, these
  34. That for any non-empty subset B of A, f (B) lies in B. The negation of the, axiom ,can thus be expressed as:: There is a set A such that for all functions f (on
  35. Of choice to be true. It is possible to prove many theorems using neither the, axiom ,of choice nor its negation; this is common in constructive mathematics. Such
  36. Seems to have gone unnoticed until Carmelo. Not every situation requires the, axiom ,of choice. For finite sets X, the axiom of choice follows from the other axiom s
  37. To define the object in the language of set theory. For example, while the, axiom ,of choice implies that there is a well-ordering of the real numbers, there are
  38. Model. The restriction to ZF renders any claim that relies on either the, axiom ,of choice or its negation unprovable. For example, the Banach–Tarski paradox is
  39. With the axiom of choice, such as the axiom of determinant. Unlike the, axiom ,of choice, these alternatives are not ordinarily proposed as axiom s for
  40. Collections, to arbitrary collections. Usage Until the late 19th century,the, axiom ,of choice was often used implicitly, although it had not yet been formally
  41. To an element of the Cartesian product of all distinct sets in the family. The, axiom ,of choice asserts the existence of such elements; it is therefore equivalent to
  42. Is a well-ordering of the real numbers, there are models of set theory with the, axiom ,of choice in which no well-ordering of the reals is definable. As another
  43. Becomes that of constructing a well-ordering, which turns out to require the, axiom ,of choice for its existence; every set can be well-ordered if and only if the
  44. Is a nonempty set. Variants There are many other equivalent statements of the, axiom ,of choice. These are equivalent in the sense that, in the presence of other
  45. Of choice, are non-measurable sets. The majority of mathematicians accept the, axiom ,of choice as a valid principle for proving new results in mathematics. The
  46. The axiom of choice, but it is consistent that no such set is definable. The, axiom ,of choice produces these intangibles (objects that are proven to exist by a
  47. All s in X." In general, it is impossible to prove that F exists without the, axiom ,of choice, but this seems to have gone unnoticed until Carmelo. Not every
  48. To the axiom of choice, and mathematicians look for results that require the, axiom ,of choice to be false, though this type of deduction is less common than the
  49. Such a selection can be obtained only by invoking the axiom of choice. The, axiom ,of choice was formulated in 1904 by Ernst Carmelo. Although originally
  50. Expect to find an algorithm to find a point in each orbit, without using the, axiom ,of choice. See non-measurable set#Example for more details. The reason that we

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