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A Functional Approach to Computability on Real Numbers
, 1993
"... The aim of this thesis is to contribute to close the gap existing between the theory of computable analysis and actual computation. In order to study computability over real numbers we use several tools peculiar to the theory of programming languages. In particular we introduce a special kind of ty ..."
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The aim of this thesis is to contribute to close the gap existing between the theory of computable analysis and actual computation. In order to study computability over real numbers we use several tools peculiar to the theory of programming languages. In particular we introduce a special kind of typed lambda calculus as an appropriate formalism for describing computations on real numbers. Furthermore we use domain theory, to give semantics to this typed lambda calculus and as a conseguence to give a notion of computability on real numbers. We discuss the adequacy of ScottDomains as domains for representing real numbers. We relate the Scott topology on such domains to the euclidean topology on IR. Domain theory turns out to be useful also in the study of higher order functions. In particular one of the most important results contained in this thesis concerns the characterisation of the topological properties of the computable higher order functions on reals. Our approach allows more...
ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...
The Logic of Brouwer and Heyting
, 2007
"... Intuitionistic logic consists of the principles of reasoning which were used informally by ..."
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Intuitionistic logic consists of the principles of reasoning which were used informally by
Various Continuity Properties in Constructive Analysis
"... . This paper deals with continuity properties of functions f : X ! Y between metric spaces X and Y within the framework of Bishop's constructive mathematics. We concentrate on the situation when X is not complete. We investigate the relations between the properties of nondiscontinuity, seque ..."
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. This paper deals with continuity properties of functions f : X ! Y between metric spaces X and Y within the framework of Bishop's constructive mathematics. We concentrate on the situation when X is not complete. We investigate the relations between the properties of nondiscontinuity, sequential continuity, mapping Cauchy sequences to totally bounded sequences, and a certain boundedness condition. 1. Introduction In constructive mathematics, investigations into conditions that ensure the continuity of a function from one metric space to another go back at least to Brouwer, who proved that every function from the real numbers to a metric space must be continuous [3, 4]. In recursive constructive mathematics, Markov showed, in 1954, that every function f : R ! R is nondiscontinuous 1 . Tsejtin [20], and Kreisel, Lacombe and Shoeneld [14] extended this to show that every function of a complete separable metric space into a separable metric space is continuous. Orevkov [16] pro...
History of constructivism in the 20th century
"... In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965, ..."
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In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965,
How Connected is the Intuitionistic Continuum?1
"... In the twenties Brouwer established the wellknown continuity theorem “every real function is locally uniformly continuous”, [Brouwer 1924, Brouwer 1924a, Brouwer 1927]. From this theorem one immediately concludes that the continuum is indecomposable (unzerlegbar), i.e. if R = A∪B and A∩B = ∅ (de ..."
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In the twenties Brouwer established the wellknown continuity theorem “every real function is locally uniformly continuous”, [Brouwer 1924, Brouwer 1924a, Brouwer 1927]. From this theorem one immediately concludes that the continuum is indecomposable (unzerlegbar), i.e. if R = A∪B and A∩B = ∅ (denoted by R = A+B), then R = A or R = B. Brouwer deduced the indecomposability directly from the fan theorem (cf. the 1927 Berline Lectures, [Brouwer 1992, p. 49]). The theorem was published for the first time in [Brouwer 1928], it was used to refute the principle of the excluded middle: ¬∀x ∈ R(x ∈ Q ∨ ¬x ∈ Q). The indecomposability of R is a peculiar feature of constructive universa, it shows that R is much more closely knit in constructive mathematics, than in classical mathematics. The classical comparable fact is the topological connectedness of R. In a way this characterizes the position of R: the only (classically) connected subsets of R are the various kinds of segments. In intuitionistic mathematics the situation is different; the continuum has, as it were, a syrupy