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Arguments for the Continuity Principle
, 2000
"... Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences ..."
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Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences . . . . . . . . . 15 3.2 Kripke's Schema and full PEM . . . . . . . . . . . . . . . . . 15 3.3 The KLST theorem . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion 19 1 The continuity principle There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the rst time in print in [Brouwer 1918]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fa
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 by providi ..."
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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 ...
ITERATED DEFINABILITY, LAWLESS SEQUENCES AND BROUWER’S CONTINUUM
"... Abstract. The research on which this article is based was motivated by the wish to find a model of Kreisel’s lawless sequence axioms in which the lawlike and lawless sequences form disjoint, inhabited, welldefined classes within Brouwer’s continuum. The original results, reported as they developed ..."
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Abstract. The research on which this article is based was motivated by the wish to find a model of Kreisel’s lawless sequence axioms in which the lawlike and lawless sequences form disjoint, inhabited, welldefined classes within Brouwer’s continuum. The original results, reported as they developed in four papers over a period of ten years from 1986 to 1996, have so far lacked a readerfriendly presentation. Since the question of absolute definability is related to the subject of these Bristol Workshops, I offer here a straightforward exposition of the final model and formal system with axioms for numbers, lawlike sequences, and arbitrary choice sequences. A choice sequence is defined to be lawless if it satisfies an extensional (un)predictability condition from which extensional versions of Kreisel’s axioms of open data and strong continuous choice follow. The law of excluded middle can be assumed for properties of lawlike and independent lawless sequences only, while Brouwer’s continuity principle applies to properties of all choice sequences. Iterating definability, quantifying over numbers and over lawlike and independent lawless sequences, yields a classical model of the lawlike sequences with a natural wellordering. Under the (classically consistent and intuitionistically plausible) assumption that the closure ordinal of the iteration is countable, a realizability interpretation establishes the consistency of a common extension FIRM(≺) of classical analysis R and Kleene’s intuitionistic analysis FIM. Lawlike sequences behave classically, while the lawless sequences form a disjoint, Baire comeager collection of choice sequences, of classical measure zero. Thus Brouwer’s continuum can be understood as a relatively chaotic expansion of a completely determined, wellordered classical continuum. 1.