Nondeducibility of the uniformization principles from Church's thesis in intuitionistic set theory

1988 ◽  
Vol 43 (5) ◽  
pp. 394-398 ◽  
Author(s):  
V. Kh. Khakhanyan
Keyword(s):  
Set Theory ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Intuitionistic Set Theory

Small decidable sheaves

1986 ◽  
Vol 51 (3) ◽  
pp. 726-731 ◽  
Author(s):  
Andreas Blass ◽  
Andre Scedrov
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
Church’S Thesis ◽  
Internal Logic ◽  
Recursive Models ◽  
Fan Theorem ◽  
Church's Thesis ◽  
Divisible Hull ◽  
Constructive Algebra ◽  
Intuitionistic Set Theory

Fred Richman conjectured that the following principle is not constructive:(*) If A is a decidable subset of the set N of natural numbers and if, for every decidable subset B of N, either A ⊆ B or A ⊆ N − B, then, for some n ∈ N, A ⊆ {n}.A set A of natural numbers is called decidable if ∀n(n ∈ A ∨ ⌉ (n ∈ A)) holds. In recursive models, this agrees with the recursion-theoretic meaning of decidability. In other contexts, “complemented” and “detachable” are often used.Richman's conjecture was motivated by the problem of uniqueness of divisible hulls of abelian groups in constructive algebra. Richman showed that a countable discrete abelian p-group G has a unique (up to isomorphism over G) divisible hull if the subgroup pG is decidable. He also showed that the converse implies.We confirm the nonconstructive nature of by showing (in §1) that it is not provable in intuitionistic set theory, IZF. Thus, in the models we construct, there are countable discrete abelian p-groups G whose divisible hulls are unique but whose subgroups pG are not decidable.Our models do not satisfy further conditions imposed by Richman, namely Church's Thesis and Markov's Principle, so the full conjecture remains an open problem. We do, however, show (in §2) how to embellish our first model so that the fan theorem (i.e., compactness of 2N) fails. (Church's Thesis implies the stronger statement that the negation of the fan theorem holds.)Our models will be constructed by the method of sheaf semantics [1], [3]. That is, we shall construct Grothendieck topoi in whose internal logic fails.

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.

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

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.

Proving Church's Thesis

2000 ◽  
Vol 8 (3) ◽  
pp. 244-258 ◽  
Author(s):  
ROBERT BLACK
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis

scholarly journals Reflections on Church's thesis.

1987 ◽  
Vol 28 (4) ◽  
pp. 490-498 ◽  
Author(s):  
Stephen C. Kleene
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis

Trial and error predicates and the solution to a problem of Mostowski

1965 ◽  
Vol 30 (1) ◽  
pp. 49-57 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Finite Number ◽  
Correct Answer ◽  
Decision Procedure ◽  
Finite Sequence ◽  
Decision Procedures ◽  
Turing Machines ◽  
Trial And Error ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Empirical Means

The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the following question: we know what sets are “decidable” — namely, the recursive sets (according to Church's Thesis). But what happens if we modify the notion of a decision procedure by (1) allowing the procedure to “change its mind” any finite number of times (in terms of Turing Machines: we visualize the machine as being given an integer (or an n-tuple of integers) as input. The machine then “prints out” a finite sequence of “yesses” and “nos”. The last “yes” or “no” is always to be the correct answer.); and (2) we give up the requirement that it be possible to tell (effectively) if the computation has terminated? I.e., if the machine has most recently printed “yes”, then we know that the integer put in as input must be in the set unless the machine is going to change its mind; but we have no procedure for telling whether the machine will change its mind or not.The sets for which there exist decision procedures in this widened sense are decidable by “empirical” means — for, if we always “posit” that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)

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