scholarly journals Constructive Mathematics, Church’s Thesis, and Free Choice Sequences

2021 ◽  
pp. 446-456
Author(s):  
D. A. Turner
Keyword(s):  
Free Choice ◽  
Constructive Mathematics ◽  
Church’S Thesis ◽  
Church's Thesis

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

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.

Can there be no nonrecursive functions?

1971 ◽  
Vol 36 (2) ◽  
pp. 309-315 ◽  
Author(s):  
Joan Rand Moschovakis
Keyword(s):  
Free Choice ◽  
Church’S Thesis ◽  
Recursive Functions ◽  
Choice Sequence ◽  
Church's Thesis ◽  
Theoretic Function ◽  
Ordinary Number ◽  
Definable Function

In 1936 Alonzo Church proposed the following thesis: Every effectively computable number-theoretic function is general recursive. The classical mathematician can easily give examples of nonrecursive functions, e.g. by diagonalizing a list of all general recursive functions. But since no such function has been found which is effectively computable, there is as yet no classical evidence against Church's Thesis.The intuitionistic mathematician, following Brouwer, recognizes at least two notions of function: the free-choice sequence (or ordinary number-theoretic function, thought of as the ever-finite but ever-extendable sequence of its values) and the sharp arrow (or effectively definable function, all of whose values can be specified in advance).

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.)

The Status of Church’s Thesis

2013 ◽  
Author(s):  
Roman Murawski ◽  
Jan Wolenski
Keyword(s):  
Church’S Thesis ◽  
The Status ◽  
Church's Thesis
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