L.E.J. Brouwer, Uber Definitionsbereiche von Funktionen, Math. Annalen 97, 1927, pp. 60-75, also in [5], pp. 390-405, see also [9], pp. 446-463. Home/Search Document Not in Database Summary Related Articles Check |
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Almost The Fan Theorem - Veldman (2001) (Correct)
....lies to the left of 1, then x itself does not lie to the right of 0. Obviously, finding such a point is beyond our means. 1. 4 Contrary to what must be by now our expectation, Brouwer succeeded in proving the Uniform Continuity Theorem, and even more, in his own intuitionistic way, see [3] and [4]. He came to understand that one must know quite a lot in order to be sure that a function from [0, 1] to R may be e#ectively calculated in every point of its domain, indeed so much that one may use this knowledge as a basis for the conclusion of the Uniform Continuity Theorem. A first indication ....
....that, for every # in X, if 2 m , then there exists # going through . #(n 1)# and really coinciding with #, and therefore p and f(#) f(#) so and also p . # 2. 6 The Continuity Theorem is due to Brouwer although its above formulation and proof are not precisely his, see [3] [4], 14] 17] Enunciating the Continuity Principle is only the first step Brouwer takes in analyzing the notion of a real function defined everywhere on R. In order to explain his next step we have to study stumps. 6 Every stump is a decidable subset of the set N # of all finite sequences of ....
L.E.J. Brouwer, Uber Definitionsbereiche von Funktionen, Math. Annalen 97 (1927), pp. 60--75 [ook in [5], pp. 390--405].
Unknown - Self-citation (Brouwer) (Correct)
....in X if for every # in X there exists n such that #n belongs to P . 1.6.6 Fan Theorem: Let F be a fan. if P is a bar in F , then some finite subset of P is a bar in F . Brouwer used the Fan Theorem for proving that every real function defined on [0, 1] is uniformly continuous on [0, 1] see [3]. 1.7 We show how one may introduce real numbers into intuitionistic analysis. 1.7.1 Let # be an enumeration of the set Q of rational numbers. # is called a real number if and only if, for each n, # , and, for every q, r in Q, if q r, then there exists n such that either ....
....coincides with an element of B and conversely, every element of B really coincides with an element of A. 1.7.3 Observe that R really coincides with Crn and that Crn, viewed as a subset of , is a spread. This fact lies at the basis of the famous result that every real function is continuous, see [3] and [22] 2 The second step, and our first one We introduce sets from the second level of the Borel Hierarchy and prove the corresponding case of the Hierarchy Theorem. 2.1 Let X be a subset of Baire space . X is basic open if and only if either X is empty or there exists s in N such that ....
L.E.J. Brouwer, Uber Definitionsbereiche von Funktionen, Math. Annalen 97, 1927, pp. 60-75, also in [5], pp. 390-405, see also [9], pp. 446-463.
Constructive Topology and Combinatorics - Coquand (1991) (4 citations) (Correct)
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Brouwer, L. E. J., Uber definitionsbereiche von Funktionen, Math. Ann. 96 (1927), 60 - 75.
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