Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WC-N . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WC-N . . . . . . . . . . . . . . . . . . 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

notions, such as `constructive proof', `arbitrary number-theoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifier-free 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 1887--1963 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 quantifier-free 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 ...