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History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...
History of constructivism in the 20th century
"... In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965, ..."
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In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965,
CARNAP, TARSKI, AND QUINE’S YEAR TOGETHER: LOGIC, MATHEMATICS, AND SCIENCE
, 2006
"... This dissertation was presented by Gregory G. FrostArnold It was defended on ..."
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This dissertation was presented by Gregory G. FrostArnold It was defended on
FINITISM = PRA? ON A THESIS OF W.W. TAIT
 REPORTS ON MATHEMATICAL LOGIC 39 (2005), 3–26
, 2005
"... ... that the chief difficulty for everyone who wishes to understand Hilbert’s conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argued that all finitist reasoning ..."
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... that the chief difficulty for everyone who wishes to understand Hilbert’s conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argued that all finitist reasoning is essentially primitive recursive. In our paper, we attempt to show that his thesis “The finitist functions are precisely the primitive recursive functions ” is disputable and that another, likewise defended by him, is untenable. The second thesis is that the finitist theorems are precisely those Π0 1sentences that can be proved in (QFIA).