Stewart Shapiro. Introduction—intensional mathematics and constructive mathematics. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, vol. 113, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 1–10. - Stewart Shapiro. Epistemic and intuitionistic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 11–46. - John Myhill. Intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 47–61. - Nicolas D. Goodman. A genuinely intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 63–79. - Andrej Ščedrov. Extending Godel's modal interpretation to type theory and set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 81–119. - Robert C. Flagg. Church's thesis is consistent with epistemic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 121–172.

1991 ◽  
Vol 56 (4) ◽  
pp. 1496-1499 ◽  
Author(s):  
Craig A. Smoryński
Keyword(s):  
New York ◽  
Set Theory ◽  
Type Theory ◽  
Foundations Of Mathematics ◽  
Constructive Mathematics ◽  
Modal Interpretation ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Epistemic Arithmetic ◽  
Intuitionistic Arithmetic

Mathematical Determinacy

2020 ◽  
pp. 239-275
Author(s):  
Jared Warren
Keyword(s):  
Set Theory ◽  
The Other ◽  
Mathematical Truth ◽  
Church’S Thesis ◽  
Subsequent Discussion ◽  
Church's Thesis ◽  
Philosophical Importance

This chapter addresses the second major challenge for the extension of conventionalism from logic to mathematics: the richness of mathematical truth. The chapter begins by distinguishing indeterminacy from pluralism and clarifying the crucial notion of open-endedness. It then critically discusses the two major strategies for securing arithmetical categoricity using open-endedness; one based on a collapse theorem, the other on a kind of anti-overspill idea. With this done, a new argument for the categoricity of arithmetic is then presented. In subsequent discussion, the philosophical importance of this categoricity result is called into question to some degree. The categoricity argument is then supplemented by an appeal to the infinitary omega rule, and an argument is given that beings like us can actually follow the omega rule without any violation of Church’s thesis. Finally, the chapter discusses the extension of this type of approach beyond arithmetic, to set theory and the rest of mathematics.

Continuity and nondiscontinuity in constructive mathematics

1991 ◽  
Vol 56 (4) ◽  
pp. 1349-1354 ◽  
Author(s):  
Hajime Ishihara
Keyword(s):  
Constructive Mathematics ◽  
Weak Version ◽  
Church’S Thesis ◽  
Continuity Properties ◽  
Church's Thesis

AbstractThe purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that “every mapping is strongly extensional”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tseĭtin theorem.

Lifschitz' realizability

1990 ◽  
Vol 55 (2) ◽  
pp. 805-821 ◽  
Author(s):  
Jaap van Oosten
Keyword(s):  
Axiomatic Characterization ◽  
Uniqueness Condition ◽  
Church’S Thesis ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Continuous Application ◽  
Church's Thesis ◽  
Order Arithmetic ◽  
Intuitionistic Arithmetic

AbstractV. Lifschitz defined in 1979 a variant of realizability which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis. In this paper we describe an extension of intuitionistic arithmetic in which the soundness of Lifschitz' realizability can be proved, and we give an axiomatic characterization of the Lifschitz-realizable formulas relative to this extension. By a “q-variant” we obtain a new derived rule. We also show how to extend Lifschitz' realizability to second-order arithmetic. Finally we describe an analogous development for elementary analysis, with partial continuous application replacing partial recursive application.

Church's thesis, continuity, and set theory

1984 ◽  
Vol 49 (2) ◽  
pp. 630-643 ◽  
Author(s):  
M. Beeson ◽  
A. Ščedrov
Keyword(s):  
Set Theory ◽  
Church’S Thesis ◽  
Church's Thesis

AbstractUnder the assumption that all “rules” are recursive (ECT) the statement Cont(NN, N) that all functions from NN to N are continuous becomes equivalent to a statement KLS in the language of arithmetic about “effective operations”. Our main result is that KLS is underivable in intuitionistic Zermelo-Fraenkel set theory + ECT. Similar results apply for functions from R to R and from 2N to N. Such results were known for weaker theories, e.g. HA and HAS. We extend not only the theorem but the method, fp-realizability, to intuitionistic ZF.

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