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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,
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, 1998
"... and Craig Evans who alerted me to the fact that the time had come to use it. Abstract. In this paper we consider a “flow ” of nonparametric solutions of the volume constrained Plateau problem with respect to a convex planar curve. Existence and regularity is obtained from standard elliptic theory, a ..."
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and Craig Evans who alerted me to the fact that the time had come to use it. Abstract. In this paper we consider a “flow ” of nonparametric solutions of the volume constrained Plateau problem with respect to a convex planar curve. Existence and regularity is obtained from standard elliptic theory, and convexity results for small volumes are obtained as an immediate consequence. Finally, the regularity is applied to show a strong stability condition (Theorem 8) for all volumes considered. This condition, in turn, allows us to adapt an argument of Cabré and Chanillo [CC97] which yields that any solution enclosing a nonzero volume has a unique nondegenerate critical point. Introduction.
Chapter 0: The Easy Way to Gödel’s Proof and Related Topics ∗
"... This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observati ..."
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This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observations on circularity and draw brief comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motives for it are made obvious. We suggest this as a more efficient way to teach the topic than what is found in the standard textbooks. For the sake of self–containment Cantor’s original diagonalization is included. A broader and more technical perspective on diagonalization is given in [Gaifman 2005]. In [1891] Cantor presented a new type of argument that shows that the set of all binary sequences (sequences of the form a0, a1,…,an,…, where each ai is either 0 or 1) is not denumerable ─ that is, cannot be arranged in a sequence, where the index ranges over the natural numbers. Let A0, A2,…An, … be a sequence of binary sequences. Say An = an,0, an,1, …, an,i, …. Define a new sequence A * = b0, b1,…,bn, … , by putting: bn = 1, if an,n = 0, bn = 0, if an,n = 1 Then, for each n, A * ≠ An, since the n th member of A * differs from the the n th member of An. Hence, A * does not appear among the Ai’s. A diagram of the following form, which appears already in Cantor’s original paper, illustrate the idea. The new sequence A * is obtained from the diagonal, by changing each of its values. The method came to be known as diagonalization. A0 = a0,0 a0,1... a0,n... A1 = a1,0 a1,1... a1,n...
Naming and Diagonalization, from Cantor to Gödel to Kleene
"... We trace selfreference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form ðD, typeð Þ, f gÞ, where D is a nonempty set; for every a 2 D, which is a name of a kary function, fag: Dk! D ..."
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We trace selfreference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form ðD, typeð Þ, f gÞ, where D is a nonempty set; for every a 2 D, which is a name of a kary function, fag: Dk! D is the function named by a, and typeðaÞ is the type of a, which tells us if a is a name and, if it is, the arity of the named function. Under quite general conditions we get a fixed point theorem, whose special cases include the fixed point theorem underlying Gödel’s proof, Kleene’s recursion theorem and many other theorems of this nature, including the solution to simultaneous fixed point equations. Partial functions are accommodated by including ‘‘undefined’ ’ values; we investigate different systems arising out of different ways of dealing with them. Manysorted naming systems are suggested as a natural approach to general computatability with many data types over arbitrary structures. The first part of the paper is a historical reconstruction of the way Gödel probably derived his proof from Cantor’s diagonalization, through the semantic version of Richard. The incompleteness proof–including the fixed point construction–result from a natural line of thought, thereby dispelling the appearance of a ‘‘magic trick’’. The analysis goes on to show how Kleene’s recursion theorem is obtained along the same lines.