Extensionality versus Constructivity
TL;DR: In this article, the authors analyze some extensions of Martin-Lof's constructive type theory by means of extensional set constructors and show that often the most natural requirements over them lead to classical logic or even to inconsistency.
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Abstract: We will analyze some extensions of Martin-Lof's constructive type theory by means of extensional set constructors and we will show that often the most natural requirements over them lead to classical logic or even to inconsistency.
read more
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Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödel’s 'Completeness Paper' (1930)
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A cartesian closed category in Martin-Löf's intuitionistic type theory
TL;DR: All the definitions and the proof of all the properties that one needs in order to show that InfBas is indeed a cartesian closed category can be formalized within Martin-Lof's intuitionistic type theory.
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The axiom of choice and the law of excluded middle in weak set theories
TL;DR: A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced and it is shown that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation of the law of excluded middle to go through.
On the formal points of the formal topology of the binary tree
TL;DR: This work analyzes one of the main concepts in formal topology, namely, the notion of formal point and contrasts two classically equivalent definitions of formal points to see that from a constructive point of view they are completely different.
The axiom of choice is false intuitionistically (in most contexts)
TL;DR: A survey of the status of the axiom of choice in various intuitionistic and constructivist systems is given in this paper , showing that it fails to be a theorem in most contexts and is even outright false in some important contexts.
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References
•Book
Programming in Martin-Lo¨f's type theory: an introduction
Bengt Nordström,Kent Petersson,Jan M. Smith +2 more
- 19 Jul 1990
TL;DR: Polymorphic sets: the semantics of the judgement forms general rules enumeration sets Cartesian product of a family of sets equality sets natural numbers lists cartesian product two sets disjoint union of two sets Disjoint Union of small sets (the first universe) well-orderings general trees.
573
Can You Add Power‐Sets to Martin‐Lof's Intuitionistic Set Theory?
TL;DR: It is shown that an extension of Martin-Lof s intensional set theory by means of a set contructor P such that the elements of P(S) are the subsets of the set S cannot be constructive.
The inconsistency of higher order extensions of Martin-Löf's type theory
TL;DR: Martin-Löf's constructive type theory forms the basis of this paper and it is shown that addition of an axiom — treating the category of propositions as a set and thereby enabling higher order quantification — leads to inconsistency.
15
Inductively generated formal topologies
TL;DR: It is shown however that some natural complete Heyting algebra cannot be inductively defined, although many formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively.