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

The word ( theorem ), is the 9002 most frequently used in English word vocabulary

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  1. Solution set condition has a left-adjoint (the Fred adjoint functor, theorem ,). Weaker forms There are several weaker statements that are not equivalent to
  2. Well-ordered. Consequently, every cardinal has an initial ordinal. **Tarski's, theorem ,: For every infinite set A, there is a bijective map between the sets A and A×A.
  3. X − a2) ··· (x − an) + 1 has no zero in F. By contrast, the fundamental, theorem ,of algebra states that the field of complex numbers is algebraically closed.
  4. Choice. The most important among them are Zorn's lemma and the well-ordering, theorem , In fact, Zermelo initially introduced the axiom of choice in order to
  5. A piece of data provided as input to a subroutine *Argument principle,a, theorem ,in complex analysis of meromorphic functions inside and on a closed contour
  6. In functional analysis, allowing the extension of linear functional **The, theorem ,that every Hilbert space has an orthonormal basis. **The Banach–Algol theorem
  7. Polynomials in FX are all degree 1,the property stated above holds by the, theorem ,on partial fraction decomposition. On the other hand, suppose that the property
  8. Boolean algebras needs the Boolean prime ideal theorem . **The Nielsen–Schrader, theorem , that every subgroup of a free group is free. **The additive groups of R and C
  9. The debate is interesting enough, however,that it is considered of note when a, theorem ,in ZFC (ZF plus AC) is logically equivalent (with just the ZF axioms) to
  10. Of a proof. It is possible, however,that there is a shorter proof of a, theorem ,from ZFC than from ZF. The axiom of choice is not the only significant
  11. To follow a multiplicative model. According to Cauchy's functional equation, theorem , the logarithm is the only continuous transformation that transforms real
  12. Consistent, Kurt Gödel showed that the negation of the axiom of choice is not a, theorem ,of ZF by constructing an inner model (the constructive universe) which
  13. To the sum of the measures of the individual sets. **Stone's representation, theorem ,for Boolean algebras needs the Boolean prime ideal theorem . **The
  14. The Cartesian product of any family of nonempty sets is nonempty. **König's, theorem ,: Colloquially, the sum of a sequence of cardinals is strictly less than the
  15. Of a set of non-logical axioms. The completeness theorem and the incompleteness, theorem , despite their names, do not contradict one another. Further discussion Early
  16. Theorem that every Hilbert space has an orthonormal basis. **The Banach–Algol, theorem ,about compactness of sets of functional. **The Bear category theorem about
  17. Sentences in a given signature is weaker, equivalent to the Boolean prime ideal, theorem ,; see the section" Weaker forms" below. Category theory There are several
  18. Polygon and b is the number of boundary points. This result is known as Pick's, theorem , Area in calculus *the area between the graphs of two functions is equal to the
  19. Mathematicians had a grasp of the principles underlying the Pythagorean, theorem , knowing, for example, that a triangle had a right angle opposite the
  20. Other choice axioms weaker than axiom of choice include the Boolean prime ideal, theorem ,and the axiom of uniformization. The former is equivalent in ZF to the
  21. For example," true in the intended interpretation ". Gödel's completeness, theorem ,establishes the completeness of a certain commonly used type of deductive
  22. Tychonoff space has a Stone–Czech compactification. **Gödel's completeness, theorem ,for first-order logic: every consistent set of first-order sentences has a
  23. About complete metric spaces, and its consequences, such as the open mapping, theorem ,and the closed graph theorem . **On every infinite-dimensional topological
  24. Developed for this purpose, to show that the axiom of choice itself is not a, theorem ,of ZF by constructing a much more complex model which satisfies AFC (ZF with
  25. And its consequences, such as the open mapping theorem and the closed graph, theorem , **On every infinite-dimensional topological vector space there is a
  26. Of a perfect crystal vanishes at absolute zero. The original Ernst heat, theorem ,makes the weaker and less controversial claim that the entropy change for any
  27. In order to formalize his proof of the well-ordering theorem . **Well-ordering, theorem ,: Every set can be well-ordered. Consequently, every cardinal has an initial
  28. Theory and they allowed Gödel to establish his famous second incompleteness, theorem , We have a language \Mather_ = \\, where 0\, is a constant symbol and S\, is
  29. Use is that a number of important mathematical results, such as Tychonoff's, theorem , require the axiom of choice for their proofs. Contemporary set theorists also
  30. The numbers \pi and e are not algebraic numbers (see the Lineman–Weierstrass, theorem ,); hence they are transcendental. * The constructive numbers (those that
  31. Of a normed vector space over the reals has an extreme point. **Tychonoff's, theorem ,stating that every product of compact topological spaces is compact. **In the
  32. A fellow at King's on the strength of a dissertation on the central limit, theorem , In 1928,German mathematician David Hilbert had called attention to the
  33. Hand that of completeness of a set of non-logical axioms. The completeness, theorem ,and the incompleteness theorem , despite their names, do not contradict one
  34. Of nonempty sets has a choice function. However, that particular case is a, theorem ,of Carmelo–Frankel set theory without the axiom of choice (ZF); it is easily
  35. The axiom of choice in order to formalize his proof of the well-ordering, theorem , **Well-ordering theorem : Every set can be well-ordered. Consequently, every
  36. Oint_^ y \dot x \, dt \point_^ (x \dot y - y \dot x) \, dt (see Green's, theorem ,): or the z-component of:: \point_^ \DEC u \times \dot \, dt. Surface area of
  37. Meaning here than it does in the context of Gödel's first incompleteness, theorem , which states that no recursive, consistent set of non-logical axioms \Sigma\
  38. Questions are given below. Roger Penrose is among those who claim that Gödel's, theorem ,limits what machines can do. (See The Emperor's New Mind. ) Martin Ford
  39. G_S in which S is a Bore subset of Bear space is determined. **The Vital, theorem ,on the existence of non-measurable sets which states that there is a subset of
  40. Theorem about compactness of sets of functional. **The Bear category, theorem ,about complete metric spaces, and its consequences, such as the open mapping
  41. Was shown to be incorrect when the Aspect experiment of 1982 confirmed Bell's, theorem , which had been promulgated in 1964. Political and religious views Albert
  42. As at absolute zero. This explains the failure of the classical equipartition, theorem ,for metals that eluded classical physicists in the late 19th century. Relation
  43. Presumably) indistinguishable from each other. " Tarsi tried to publish his, theorem ,the equivalence between AC and 'every infinite set A has the same cardinality
  44. S representation theorem for Boolean algebras needs the Boolean prime ideal, theorem , **The Nielsen–Schrader theorem , that every subgroup of a free group is free.
  45. Of the absolute value for real numbers, it follows from the Pythagorean, theorem ,that the absolute value of a complex number is the distance in the complex
  46. Enforrce determinism. However, thirty years later, in 1964,John Bell found a, theorem ,involving complicated optical correlations (see Bell inequalities),which
  47. Is free. **The additive groups of R and C are isomorphic. And **The Hahn–Banach, theorem ,in functional analysis, allowing the extension of linear functional **The
  48. Surely) and write: (\formally\nifty n) P (n). For example, the prime number, theorem ,states that the number of prime numbers less than or equal to N is
  49. This is a result of Galois theory (see Quin tic equations and the Abel–Roughing, theorem ,). An example of such a number is the unique real root of polynomial (which is
  50. Uniquely determining its structure. In fact the Deaconess–Goodman–My hill, theorem ,shows how to derive the constructively unacceptable law of the excluded middle

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