E. Bishop and D. Bridges, Constructive Analysis, Springer-Verlag, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the influence of Brouwer s ideas, distrusted negation and emphasized the importance of so called a#rmative mathematics. 0.5 It is possible to prove the Hierarchy Theorem for positively Borel sets Surprisingly, this question is ignored by P. Martin Lof in [14] and by E. Bishop and D. Bridges in [1], although also these authors seem to hold the opinion that, in constructive mathematics, the best Borel sets are positively Borel sets. Brouwer, much earlier studying positively Borel sets of the second level only, sought and found a countable union of closed sets that is not a countable ....

....Principle plays a crucial role in the proof of most of the results of this paper. Once we agree to accept and use it we enter a new world and discover many facts for which there does not exist a classical counterpart. The principle entails for instance that the union of the two closed sets [0, 1] and [1, 2] is not a countable intersection of open subsets of R. One may also infer that there are unions of three closed sets di#erent from every union of two closed sets. These observations are the tip of an iceberg. The intuitionistic Borel Hierarchy shows o# an exquisite fine structure. 0.7 ....

[Article contains additional citation context not shown here]

E. Bishop and D. Bridges, Constructive Analysis, Springer-Verlag, Berlin, 1985.

....Theorem. Brouwer s Fan Theorem is a bone of contention, even among constructivists. E. Bishop fullysubscribed to Brouwer s critique and agreed with him that large parts of mathematics are in need of revision, but he could not understand Brouwer s proposals for new asiomx. In [ and [2] he defends a straightforwardly realistic constructive mathematics, without semi mystical elements. In [10] S.C. Kleene showed that the Fan Theorem is incompatible with an algorithmic conception of the continuum. There exists a Turing computable subset B of N # such that every ....

E.A. Bishop and D.S. Bridges, Constructive analysis, Springer, Heidelberg & New York, 1985.

....a common re nement, which is stated without proof in [4] Unfortunately, though classically this is a trivial statement, constructively it is not true The reason for that is that in a partition points must be ordered, and is not decidable on the real numbers. This error was corrected in [5] in the following way: rst, we say that two partitions P = a 0 ; an ) and R = b 0 ; b m ) are separated i for all i and j in the appropriate ranges a i a i 1 and b j b j 1 ; furthermore, if 0 i n and 0 j m then a i 6= b j . Now, we can prove that any two separated ....

....error in Bishop s original proof was to restrict our attention to even partitions. Unfortunately, for the next result (which is a fundamental theorem, and not just an auxiliary lemma) we really need the general de nition, and at this stage we had to go back and redo our work according to [5]. We want to show that Z c f(x)dx c f(x)dx (2) whenever a c b. This is trivially done using properties of limits, closely following Bishop s proof, and appealing to (1) This requires choosing arbitrary (even) partitions of [a; b] and [b; c] and obtaining from those a partition of [a; ....

Bishop, E. and Bridges, D., Constructive Analysis, Springer-Verlag, 1985

....effort. Fortunately, nullsets bearing a strong resemblance to total recursive sequential tests were already known in constructive mathematics, so we can draw upon the large reservoir of proof techniques developed there (see, e.g. the books by Bishop [5] Bridges [9] and Bishop Bridges [6]) Although not every total recursive function is acceptable in constructive mathematics (since the proof that the function is in fact total must itself be constructively valid) arguments involving constructive functions usually carry over directly to recursive functions; when the result is simple ....

....obvious and the second statement follows since B is a G # set which is dense by = 1. In what follows, we shall often refer to computable real valued functions on 2 , the recursiontheoretic analogue of the continuous real valued functions of constructive analysis (see e.g. Bishop Bridges [6,38]) We therefore introduce 4.4.3 Definition f: 2 # is computable if it is recursively uniformly continuous, i.e. if for some total recursive h: # # : for all n, for all x,y: if x y h(n) then f(x) f(y) 2 . The first part of lemma 4.4.1 implies that if # is a computable ....

[Article contains additional citation context not shown here]

E. Bishop, D.S. Bridges, Constructive analysis, Springer-Verlag (1985).

....Research Foundation. 1 the currently common name of weak Markov s principle it has been investigated by Ishihara ( 8] 9] WMP plays a crucial role in the study of the interrelations between various continuity principles within the framework of Bishop style constructive mathematics ( 2] [3], 4] In order to state WMP we first need the notion of pseudo positivity : Definition 1 1) A real number a # IR is pseudo positive if #x # IR( 0 x) # (x a) 2) a # IR is positive if a 0. Remark 2 1) It is clear that we can without loss of generality restrict x # IR in the ....

Bishop, E., Bridges, D.S., Constructive Analysis. Springer Verlag, Berlin (1985).

....for all a 2 A such that the error of a is less or equal , the error of f( a) is less or equal . Note that this de nition corresponds to the de nition of uniform continuity in analysis; this notion and similar ones are used as necessary conditions for computability in e ective analysis [6, 7, 34, 45]. Observe that every element a 2 A gives us the information that the result of some computation is in a certain subset of A. In the optimal case we can identify any subset of A by an element in A. So we use subsets of A for modeling approximate computation from now on and simply identify ....

E. Bishop and D. Bridges. Constructive Analysis. Springer, 1985.

.... : x ( y) x y) x y) 0 z) x z y z) x y) z 0) y z x z) 0 x y) x # 0) y # 0) 4 Consistency and completeness The usual way for de ning real numbers is the use of (equivalence classes of) Cauchy sequences of rational numbers [Bee85, BB85, TvD88] but there exist other constructions that can be easily proved equivalent to this one. An example is the approach that introduces the reals as in nite sequences of digits [PEE97, Wei00, CDG00] in this case the equivalence follows from the fact that it is possible to transform e ectively ....

E. Bishop and D. Bridges. "Constructive Analysis". Springer-Verlag, 1985.

....in the literature constructive reals are normally de ned starting from the rational numbers and then the formal properties of these representations are carefully analyzed. 9 The usual way for de ning real numbers is the use of (equivalence classes of) Cauchy sequences of rational numbers [Bee85, BB85, TvD88] but there exist other constructions that can be easily proved equivalent to this one. An example is the construction that introduces the reals as in nite sequences of digits [PEE97, Wei00, CDG00] in this case the equivalence follows from the fact that it is possible to transform ....

E. Bishop and D. Bridges. "Constructive Analysis". Springer-Verlag, 1985.

....others. We have essentially adopted Weihrauch s approach [10] the so called Type 2 Theory of E#ectivity, which allows to express computations with real number, continuous functions and subsets in a highly uniform way. # Work partially supported by DFG Grant Me872 7 3 In constructive analysis [1], as well as in computable analysis closed subsets are often represented by distance functions, which, roughly speaking, play the role of continuous substitutes for characteristic functions (cf. 2] for a survey) Such representations of sets by distance functions can also be considered for ....

Errett Bishop and Douglas S. Bridges. Constructive Analysis, Springer, Berlin, 1985.

....be easier to start with the second hypothesis. The first hypothesis is covered in the next section. In order to formalize the second hypothesis, we must recall some basic definitions of computable ( constructive ) real numbers and computable functions from real numbers to real numbers (see, e.g. [1, 3, 4, 5, 23]) Definition 1. A real number x is called computable if there exists an algorithm (program) that transforms an arbitrary integer k into a rational number x k that is 2 Gammak Gammaclose to x. It is said that this algorithm computes the real number x. When we say that a computable real ....

....if we had an algorithm which checks, given B, f , and C, whether max f C or not, then we will be able 30 to check whether C 0 for a given computable real number C. However, it is known that it is algorithmically impossible to check whether a given computable real number is positive or not [1, 3, 4, 5, 15, 23]) Thus, a CRT algorithm cannot be always applicable. The proposition is proven. 9.2 Proof of Proposition 2 1. It is known that there exists an algorithm which, given a computable function on a computable box, and a given ffi 0 returns a rational number M which is ffi close to max f [1, 3, 4, ....

[Article contains additional citation context not shown here]

E. Bishop and D. S. Bridges, Constructive Analysis, Springer, N.Y., 1985.

.... when the intervals (from which we are choosing the simplest numbers) come from computations, it is reasonable not to consider arbitrary definable real numbers, but to restrict ourselves to computable real numbers, i.e. real numbers that can be computed with an arbitrary accuracy (see, e.g. [4, 7, 2, 5]) 9 Definition 10. A real number x is called constructive if there exists an algorithm (program) that transforms an arbitrary integer k into a rational number x k that is 2 Gammak Gammaclose to x. It is said that this algorithm computes the real number x. Comment. Every constructive real ....

E. Bishop, D. S. Bridges, Constructive Analysis, Springer, N.Y., 1985.

....within Bishop s constructive mathematics. In particular, we present a set of axioms that abstracts the constructive properties of the lattices of subspaces and projections on a Hilbert space. 1. INTRODUCTION Our discussion takes place in the context of Bishop s constructive mathematics (BISH; [3, 4]) in which existence is interpreted strictly as constructibility. 3 One distinctive feature of BISH, compared with other varieties of constructive mathematics is that its results and proofs can be interpreted mutatis mutandis within classical (that is, traditional) mathematics, recursive ....

E. Bishop and D. Bridges, Constructive Analysis, Springer-Verlag, Heidelberg, 1985.

No context found.

E. Bishop & D. Bridges, Constructive Analysis, Berlin (Springer), 1985

No context found.

E. Bishop and D. Bridges, Constructive Analysis, Springer-Verlag, 1985.

No context found.

E. Bishop and D. Bridges, Constructive Analysis, Springer-Verlag, 1985.

No context found.

E. Bishop and D. Bridges. Constructive Analysis. Springer-Verlag, 1985.

No context found.

E. Bishop, D. S. Bridges, Constructive Analysis, Springer, N.Y., 1985.

No context found.

E. Bishop and D. S. Bridges, Constructive Analysis, Springer-Verlag, New York, 1985.

No context found.

E. Bishop and D. Bridges. Constructive Analysis. Springer, New York, 1985.

No context found.

E. Bishop, D. S. Bridges, Constructive Analysis, Springer, N.Y., 1985.

No context found.

Errett Bishop and Douglas Bridges. Constructive Analysis. Springer, Berlin, 1985.

No context found.

E. Bishop and D. Bridges. Constructive Analysis. Springer-Verlag, Berlin, 1985.

No context found.

E. Bishop and D.S. Bridges (1985). Constructive Analysis. Springer.

No context found.

Bishop, E. and Bridges, D.S. (1985). Constructive Analysis. Springer.

No context found.

E. Bishop and D. Bridges, Constructive Analysis (Springer Verlag, 1985).

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC