Troelstra, A.S., D. van Dalen, Constructivism in Mathematics - An Introduction, vol. I, II, Studies in Logic and the Foundations of Mathematics, vol. 121, North Holland 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....topics, we suggest consulting Girard, Lafont, Taylor [9] or Gallier [6] for background on logic, and Barendregt [2] Hindley and Seldin [15] or Krivine [19] for background on the lambda calculus. For an in depth study of constructivism in mathematics, we highly recommend Troelstra and van Dalen [32]. 2 Natural Deduction, Simply Typed Calculus We rst consider a syntactic variant of the natural deduction system for implicational propositions due to Gentzen [8] and Prawitz [23] In the natural deduction system of Gentzen and Prawitz, a deduction consists in deriving a proposition from a ....

A.S. Troelstra and D. van Dalen. Constructivism in Mathematics: An Introduction, Vol. I and II, volume 123 of Studies in Logic. North-Holland, 1988.

....recursive characteristic function #R . Intuitionistically, only those relations which are continuous in their choice sequence variables can have total characteristic functions (which may or may The Friedman Dragalin translation, in conjunction with Kleene s Rule, justifies MR in each case; see [19] or [1] for details. Translating by Q proves that these theories, as well as the subtheory of without the axiom MPPR , also satisfy the independence of premise rule IPR1 : If A # ##B(#) holds, where A does not depend on #, then ##(A # B(#) holds. Since it conflicts with IPR1 , MPPR is ....

A. S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction, 1 and 2 (North-Holland, Amsterdam, 1988).

....of constructive analysis is needed in order to understand the work below: an awareness of the di#erences between classical and intuitionistic logic should su#ce. However, the reader may benefit from keeping at hand either [3] or [6] Other general references for constructive mathematics are [2, 10, 20]; for the recursive approach to constructive mathematics see [1, 16] and for intuitionistic mathematics see [14, 20] # Department of Mathematics Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand, e mail: Luminita Math.net Department of Mathematics ....

....classical and intuitionistic logic should su#ce. However, the reader may benefit from keeping at hand either [3] or [6] Other general references for constructive mathematics are [2, 10, 20] for the recursive approach to constructive mathematics see [1, 16] and for intuitionistic mathematics see [14, 20]. # Department of Mathematics Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand, e mail: Luminita Math.net Department of Mathematics Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand, e mail: d.bridges math.canterbury.ac.nz ....

A.S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction (two volumes), North Holland, Amsterdam, 1988.

....Theorem 3.35 tells us that Phi ctr , which is what we concluded in Example 3.31. And vice versa. We believe that the above development would go through, mutatis mutandis, for Henkin models [Hen50] as well as in a constructive framework like that of topos theory [Pho92] see Section 3. 9 of [TvD88]. In the absence of the axiom of choice, e.g. in topos theory, one must replace the function in the proof of Theorem 3.35 by a relation which is functional up to . 18 4 Behavioural equivalence and indistinguishability We now consider specific definitions of indistinguishability and behavioural ....

....contain only first order constants, function types can be useful in formulae as shown by Example 7.6. Such types can be encoded using predicate types as shown there. We believe that the passage from the informal presentation to its encoding can be made more systematic, following ideas in [TvD88]; it also seems to be possible to encode uses of a description operator like Hilbert s which selects values that satisfy a predicate. The interpretation we give to predicate types means that this encoding yields the full set theoretic function space. This is in contrast to the encoding above, ....

A. Troelstra and D. van Dalen. Constructivism in Mathematics: An Introduction, Vol. 1. North-Holland (1988).

....exposition should serve more as a reminder and a statement about notation than a first introduction to the concepts just listed. For that a much longer text is needed, and we refer the reader to the following list of core references. Fourman Scott 77] Mac Lane Moerdijk 92] Johnstone 77] Troelstra van Dalen 88] Wyler 91] Fourman 74] and also [Fourman 77] Rosolini 80] Ambler 92] and [Nawaz 85] The intention behind the notion of an m set is to model sets in a constructive universe with truth values in . Thus operations like equality (between members of sets) and set membership that usually ....

....we want to formulate properties of apos and their elements externally, only using the global sections and the lattice operations on or internally, using logical connectives instead. We use the logic of forcing over , that is, the logic in h( For a good reference to forcing over a cHa, see [Troelstra van Dalen 88] volume II, and also in general for internal logic in a topos, for instance [Mac Lane Moerdijk 92] Johnstone 77] or [Fourman 74] 69 Definition 4.18 For an atomic formula with value I ] P c [2 we define q k if q p. For composite formulae we define forcing as follows (taken from ....

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Troelstra, A.S., D. van Dalen, Constructivism in Mathemat- ics - An Introduction, vol. I, II, Studies in Logic and the Foundations of Mathematics, vol. 121, North Holland 1988.

....be an equivalence relation that respects the functions and relations. The corresponding semantic notion for intuitionistic first order logic is that of a Kripke model. We briefly review the definition of Kripke structures as models for intuitionistic logic; see Troelstra and van Dalen s textbook [4] for a thorough treatment. Fix a first order language . A Kripke structure C for the language is an ordered pair ( gd ez , 4) where gd e7 is a set of (not necessarily distinct) classical structures for the language indexed by elements of the set 27 and where , is a reflexive and transitive ....

....2 show that T is the intuitionistic theory of T normal Kripke structures. The proof of Theorem 2 takes the rest of this section and will proceed along the lines of the proof of the usual strong completeness theorem for intuitionistic logic as exposired in section 2. 6 of Troelstra and van Dalen [4]. The new ingredient and the most difficult part in our proof is Lemma 4 which ensures that the Kripke structure we construct is T normal. We let T be a fixed classical theory for the rest of this section. For simplicity, we assume that T and its language are countable; however, the proof can be ....

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A. S. TROELSTRA AND D. VAN DALEN, Constructivism in Mathematics: An Introduction, vol. I, North-Holland, 1988.

....least upper bound of the set. Likewise, the meet V S would be its greatest lower bound. The meet and join serve to model conjunction and disjunction, but we need an additional operator corresponding to implication. The following de nition of a Heyting (or pseudo Boolean) algebra is taken from [21]. De nition 3.2 A Heyting algebra H is a lattice with a least element and an operation de ned on all pairs of elements of H such that a b c if and only if a b c (9) In a complete Heyting algebra, arbitrary meets and joins exist. We will use Heyting algebras with certain arbitrary ....

....an operation de ned on all pairs of elements of H such that a b c if and only if a b c (9) In a complete Heyting algebra, arbitrary meets and joins exist. We will use Heyting algebras with certain arbitrary meets and joins corresponding to universal and existential quanti cation. In [21] we also learn that a Heyting algebra is distributive and that if W B exists then a B = fa b : b 2 Bg (10) It is clear from (4) and (9) that a Heyting algebra has a top element, which we denote . Example 3.3 The open subsets of R form a Heyting algebra with denoting union, the ....

A. S. Troelstra and D. van Dalen. Constructivism in Mathematics: An Introduction, volume 2. Elsevier Science Publishers, 1988.

....P 1 = P 3 which contradicts the earlier finding that P 1 6= P 3 , so these coincidence and apartness relations are not consistent. Note that this inconsistency can occur with any precision, as long as there is some error. The problem demonstrated in the above example is actually more general. In[35] it is shown that equality for real numbers is not decidable with constructive means. Coincidence of geometric objects defined in the real coordinate space is therefore undecidable, as well. Introducing a third relation ambiguity besides coincidence, and apartness for situations such as in figure ....

....as well. Introducing a third relation ambiguity besides coincidence, and apartness for situations such as in figure 2 will make it possible to obtain a consistent definition of geometric relationships. A theoretical explanation of this three valued relationship is the so called intuitionistic logic[35], in which the refutation of apartness is not equivalent to equality and therefore the law of the excluded middle does not hold for equality of real numbers. The mathematics based on intuitionistic logic is also called constructive mathematics, meaning that an explicit constructive approach ....

Troelstra, A. S. Constructivism in Mathematics : An Introduction. Elsevier Science Pub. Co., 1988.

....to GnA i in [10] it is also possible to extract such majorizing terms directly from a given proof, i.e. without extracting t at first. However the simplification achieved in this way is not as significant as for the functional interpretation since no decision of prime formulas is needed 1 In [17] mrt is denoted by mq . But note that in [14] mq denotes a slightly different interpretation. 2 This variant has the property that x mrt A implies A; see [17] 16] for information on this. 6 for the mr interpretation of intuitionistic logic (in contrast to usual functional ....

....achieved in this way is not as significant as for the functional interpretation since no decision of prime formulas is needed 1 In [17] mrt is denoted by mq . But note that in [14] mq denotes a slightly different interpretation. 2 This variant has the property that x mrt A implies A; see [17], 16] for information on this. 6 for the mr interpretation of intuitionistic logic (in contrast to usual functional interpretation, where this is avoided only by our monotone variant) and it will be therefore not studied further. The monotone mr interpretation has the same nice behaviour ....

Troelstra, A.S. -- van Dalen, D., Constructivism in mathematics: An introduction. Vol. I,II. North--Holland, Amsterdam (1988). 21

....real numbers. Our main contribution is a new system of axioms, synthesized with the aim of being minimal, i.e. of assuming the least number of primitive notions and properties. Such a system is consistent with respect to reference models we have in mind (equivalence classes of) Cauchy sequences [TvD88] and co inductive streams of digits [CDG00] and will be compared to other proposals in the literature [Bri99, GPWZ00] The expressive power of the axioms will be addressed in order to guarantee that a sucient part of the constructive analysis can be derived from them. We de ne constructive real ....

....other typical properties. We have thought that it is cleaner to de ne only an order relation and to pick out a minimal set of constructive properties for it: in fact the order is universally considered the most fundamental relation for real numbers both in classical and constructive mathematics [TvD88, Bri99] The alternative would have been to start from the apartness relation [GPWZ00] i.e. the constructive non equality then to assume axioms for it and further to introduce the order itself with its proper axioms. But this increases the lenght of the presentation of the constructive reals ....

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A.S. Troelstra and D. van Dalen. "Constructivism in Mathematics, an Introduction". North-Holland, 1988.

....of constructive real numbers as 2 k in (1) could be replaced by 10 k or 1 k , k 0. Another common variant (used in [1] and [4] is x m x n # 1 m 1 n for m,n # Z . The definition stated here is however more convenient for computation purposes and is found in [16]. By taking m = p(k) and letting n # # in (1) it is seen that x p(k) is an approximation to the real number x within 2 k . Hence x p(k) is often called the kth approximation of x. As there are many di#erent fundamental sequences for the same real number, an equivalence relation for equality ....

....u : N # N such that for all k # N and all x # I, m,n # u(k) # f m (x) f n (x) # 2 k . 3) Theorem 3 A sequence f n n#N of continuous functions on a compact interval I converges to a continuous function on I if and only if it is a Cauchy Sequence. Proof: See [4] or [16]. # Hence a transcendental continuous function f on I can be defined as f = f n n#N , v n n#N , u) where for each n # N, f n : I #Q # Q is a continuous function and v n : N # N is a modulus of continuity for f n . Also, u : N # N has the property stated in (3) so that ....

A. S. Troelstra and D. van Dalen. Constructivism in Mathematics: An introduction, vol. 1. North-Holland Publ. Co., Amsterdam, 1988.

....with a definedness predicate which is usually written as t # or E(t) Logics of partial terms and precursors of it have been used for the foundation of explicit and Appeared in: Journal of Functional Programming, 8(2) 97 129, 1998. 1 constructive mathematics [4, 1] Troelstra and van Dalen [28] compare the logic of partial terms with the logic of existence [25] The Russian constructivist school of N. A. Shanin used similar logics [22] The second view of partial functions leads to D. Scott s Logic for Computable Functions (LCF) which includes in its language constants # [9] Di#erent ....

A. S. Troelstra and D. van Dalen. Constructivism in Mathematics: an Introduction. North-Holland, Amsterdam, 1988.

....semantical explanation. A similar kind of explanation can also be found in the Br#C wer# kFCxIIj Kolmogor# v inter#5jCxI .5 of fundamental logical connectives; indeed,p oofs of the pr#C ositions builtfr#l such connectives ar# semantically explained inter#k of thepr#j5 .C e notion, methods. See [45],for example. Consequently, theop enendedness of p oofs follows. Inbr#j. the open endednesspr# er# y in ITT is what enables ITT toincor# or#or new objects and types. This inter# j4C pr## er# y, however# must be expr# kCx in amor# specific andclear#5 under# jCx 4Fj for# so that it may be used ....

A. S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction, Volumes I and II (North-Holland, Amsterdam, 1988).

....to be very careful. The following substitution property that says that lambda abstraction and substitution can be commuted is easily violated. # # x.a) b y] # # # x. a[b y] for x ## y and x # FV(b) #) This property is taken for granted by most authors that use the framework of CL p (cf. [2,4,13]) To avoid the substitution problem we introduce a call by value logic LV. The terms of LV are # terms and not the combinator terms of CL p . The essential di#erence to CL p is that in LV quantifiers may only be instantiated to value terms. This means that instead of the quantifier axiom #x#(x) ....

A. S. Troelstra and D. van Dalen. Constructivism in Mathematics: an Introduction. North-Holland, Amsterdam, 1988.

....logic variables. We will discuss these features in more detail, below. To produce a suitable semantics, the authors have drawn on extensive work in the semantics of higher order intuitionistic systems by e.g. Dragalin, Scott and Fourman (Omega sets and sheaves) 14] Troelstra and van Dalen [15] and Mitchell Moggi [11] 2 The Calculus Church s calculus consists of a collection of higher order functional types together with a simply typed lambda calculus of terms. Connectives and quantifiers are explicit typed constants in the calculus which permits the formalization of full ....

....models to model equational reasoning in the typed lambda calculus. We use a similar approach to combine applicative structures with Heyting algebra semantics [15, Chap. 13, x4] for constructive logic. We assume familiarity with the definitions of lattice and (Complete) Heyting Algebra. See e.g. [15] for details. 6 3.1 Applicative Structures We will make use of the notion of applicative structures, a well known semantical framework for the simply typed lambda calculus. See, for example, 10] Definition 3.1 A typed applicative structure hD; App; Consti consists of a set D ff for each ....

A. S. Troelstra and D. van Dalen. Constructivism in Mathematics: An Introduction, volume II. North Holland, the Netherlands, 1988.

....functions and models, as these terms are used within mathematical logic [34, 55] and we are attempting this in [43] 16 However, even pure mathematicians may have variable belief in an assertion depending upon the means used to prove it. For example, constructivist mathematicians (e.g. [6, 70]) do not accept inference based on proof techniques which purport to demonstrate the existence of a mathematical object without also constructing it. Typically, such proofs use a reductio ad absurdum argument, showing that an assumption of non existence of the object leads to a contradiction. ....

A. S. Troelstra and D. van Dalen. Constructivism in Mathematics: An Introduction (Two Volumes). North-Holland, Amsterdam, The Netherlands, 1988.

....one risks using the very rule being justified. So how can one person convince another of the validity of a rule of deductive inference That rules of inference may be the subject of fierce argument is shown by the debate over Constructivism in pure mathematics in the twentieth century [21]: here the rule of inference being contested was double negation elimination in a Reductio Ad Absurdum (RAA) proof: FROM ( P Q) and ( P :Q) INFER : P FROM : P INFER P Although the choice of inference rules in purely formal mathematics may be arbitrary, 1 the question of acceptability of ....

A. S. Troelstra and D. van Dalen. Constructivism in Mathematics: An Introduction. North-Holland, Amsterdam, The Netherlands, 1988. 8

....accessibility relation) are related by two confluence relations. Kripke frames for PLL and PLL 1 are unusual for two reasons. First of all they allow fallible worlds, that is worlds in which inconsistent information is admitted (see [Cur52] or the discussion on fallible Beth models in [TvD88]) Secondly, no confluence relation between R and is required. In Section 3 we give a duplication free tableau calculus for PLL 1 and a calculus for PLL in which only one rule duplicates the signed formula to which it is applied. In Section 4 we discuss the soundness theorem for these calculi ....

A.S. Troelstra and D. van Dalen. Constructivism in mathematics: an introduction. Volume 2. In Studies in logic and the foundations of mathematics, volume 122. North--Holland, 1988. 17

....[13] for references) which implies that zf is consistent iff nbg is consistent. However, nbg is finitely axiomatizable, while zf is not (see [11] or [13] for a proof) stmm admits undefined terms and has the same kind of machinery for reasoning with functions and for classifying terms as lutins [4, 5, 6], the logic of the imps Interactive Mathematical Proof System [10] lutins closely corresponds to mathematics practice and has proven to be an effective logic for formalizing traditional mathematics (e.g. see [9] In particular, stmm is equipped with operators for forming definite ....

....CP 91 5402 43, ORA Corporation, 1991. 4] W. M. Farmer. A partial functions version of Church s simple theory of types. Journal of Symbolic Logic, 55:1269 91, 1990. 5] W. M. Farmer. A simple type theory with partial functions and subtypes. Annals of Pure and Applied Logic, 64:211 240, 1993. [6] W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al. editor, Higher Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96 123. Springer Verlag, 1994. 7] W. M. Farmer. Reasoning about partial functions with the aid of a ....

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Troelstra, A.S. and D. van Dalen (1988), Constructivism in Mathematics -- An Introduction, Volume I, North-Holland, Amsterdam. 107

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Troelstra, A.S., D. van Dalen, Constructivism in Mathematics - An Introduction, vol. I, II, Studies in Logic and the Foundations of Mathematics, vol. 121, North Holland 1988.

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Troelstra, A. S. and van Dalen, D.: 1988. Constructivism in Mathematics: An Introduction. number 121 and 123 in Studies in Logic and the Foundations of Mathematics. Elsevier. Two volumes. Valentini, S.: 2001. Fixed points of continuous functions between formal spaces. Technical report. Department of Pure and Applied Mathematics, University of Padua.

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A. S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction, vol. I, North-Holland, 1988.

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A.S. Troelstra and D. van Dalen. Constructivism in Mathematics: An Introduction, volume 2. North Holland, 1988.

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A. S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction, vol. I, North-Holland, 1988.

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A. S. Troelstra and Dirk van Dalen. Constructivism in Mathematics: An Introduction, volume 1. North-Holland, Amsterdam, 1988.

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