Hatcher, W. S., The Logical Foundations of Mathematics, Toronto, Pergamon Press, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a precise formulation of the syntax. Godel calls the vague syntax in Principia a considerable step backwards as compared with Frege [13, page 448] Church formalizes syntax, including quantifiers, in the typed # calculus. His technique is now standard in generic theorem proving. See Hatcher [16] and Godel [13] for further discussion of the history of these type theories, and Andrews [1] for the formal development. 4 Fundamental issues in type theory The following sections discuss basic issues in type theory: subtypes, description operators, empty types, polymorphism, and higher order ....

William S. Hatcher. The Logical Foundations of Mathematics. Pergammon, 1982.

....le situazioni pi u svariate, ed espressivo, ossia capace di rappresentare qualunque situazione ad un qualsiasi livello di astrazione. Attualmente i formalismi utilizzati sono molteplici ma possono essere raggruppati come segue: Formalismi basati sulla Logica. Logica del primo ordine [Hat82, Men64] Logica di Boyer Moore [BM84, BM88] Logica di ordine superiore [Rei83, Gor87] Logica Temporale [Eme90, RU71] Logica Temporale Lineare [MP82] Logica Temporale ad Intervalli [HKP82] Mu Calcolo [Koz83] con molteplici varianti. Formalismi basati sulla Teoria degli Automi. ....

W. S. Hatcher. The Logical Foundations of Mathematics. Pergamon Press, Oxford, 1982.

....as sets of pairs. In simple type theory (i.e. higher order logic) functions are primary, predicates are functions with range type boolean, and sets are identi ed with their characteristic predicates. Here, the antinomies are avoided by type rules that prohibit the vicious circle constructions [14]. The problems with axiomatic set theory as a foundation for a speci cation language are rstly that strict typechecking has to be grafted on 5 and this is not straightforward and secondly that functions are essentially partial. Partial functions further complicate strict typechecking, and are ....

William S. Hatcher. The Logical Foundations of Mathematics. Pergamon Press, Oxford, UK, 1982.

....called type, as an abbreviation for type expression. In mathematics, types impose constraints that help to avoid paradoxes. Untyped universes such as that of usual set theory have logical inconsistencies (such as Russel s Paradox) that are avoided, for example, with a typed set theory [Qui69, Hat82] In computer science, there are several untyped languages (i.e. languages that have just one type, which contains all values) such as, for example: LISP, calculus, Self, Perl, and Tcl. These languages do not have any mechanism to detect errors due to operations being applied to improper ....

W. S. Hatcher. The Logical Foundations of Mathematics. Pergamon, 1982.

....power gained by allowing higher order variables is dangerous. Consider the predicate P de ned by: P x = x x) from this de nition it follows that: P P = P P) which is a version of Russell s paradox. Russell invented a method for preventing such inconsistencies based on the use of types [Hatcher] HOL uses a simpli cation of Russell s type system due to Church [Church] with extensions developed by Milner [Milner (78) Types are expressions that denote sets of values, they are either atomic or compound . Examples of atomic types are: bool; ind; num; real these denote the sets of ....

....1 x n : t can be written as: f x 1 x n = t The HOL system currently permits the user to postulate arbitrary axioms when he builds a theory. This freedom is dangerous because inconsistent axioms can be introduced (e.g. by postulating T = F) As was shown by Russell and Whitehead [Hatcher] with suitable de nitions, all of classical mathematics can be constructed from logic together with the assumption that there are in nitely many individuals (the Axiom of In nity) It would thus appear reasonable to restrict the user to only making de nitions and we eventually plan to do this. ....

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W. Hatcher. The Logical Foundations of Mathematics. Pergamon Press, 1982. 29

....accounts of mathematical practice in certain areas, including category theory itself, and moreover have encouraged research into areas that have little or nothing to do with mathematical practice, such as large cardinals. Mac Lane [48] gives a lively discussion of these issues; see also [37] for an overview of various approaches to foundations. In any case, attempts to find a minimal set of least debatable concepts upon which to erect mathematics have little direct relevance to computing science. Of course, the issue no longer seems as urgent as it once did, because no new paradoxes ....

William S. Hatcher. The Logical Foundations of Mathematics. Permagon, 1982.

....not be well founded, see section 3.7. No problems occur however, if A has another specifier C, which is identifiable. Note that B Gen A is not possible, as Gen is a relation over A Theta O, while B 2 G. The problem of structural identification can also be related to predictive type theory (see [13]) Considering figure 27 it is clear that instances from object type B cannot be properly typed. Their types would have to be cyclic structures, while predictive type theory enforces types to be hierarchical structures. In some cases it may be difficult to establish whether object types fulfil the ....

W.S. Hatcher. The Logical Foundations of Mathematics. Pergamon Press, 1982.

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Hatcher, W. S., The Logical Foundations of Mathematics, Toronto, Pergamon Press, 1982.

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Hatcher, W. S. [1982] The Logical Foundations of Mathematics, Pergamon Press, Oxford.

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