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).

For Helmut Schwichtenberg, with respect and appreciation for his encouragement and friendship.In [2] J. Berger and Ishihara proved, via a circle of informal implications involving countable choice, that Brouwer's Fan Theorem for detachable bars on the binary fan is equivalent in Bishop's sense to various principles including a version WKL! of Weak König's Lemma with a strong effective uniqueness hypothesis. Schwichtenberg [9] proved the equivalence directly and formalized his proof in Minlog. We verify that his result does not require countable choice, and derive a separation principle SP from the Fan Theorem, in a minimal intuitionistic system M of analysis with function comprehension.In contrast, WKL!! comes from Weak König's lemma WKL by adding the hypothesis that any two infinite paths must agree. WKL!! is interderivable over M with the conjunction of a consequence of Markov's Principle and the double negation of WKL. This decomposition is in the spirit of Ishihara's [4] and J. Berger's [1]. Kleene's function realizability and the author's modified realizability establish that WKL!! is strictly weaker than WKL and strictly stronger than WKL!.

This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(α) on Baire space with the property that every constructive partial functional defined on {α: R(α)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(α) is equivalent in to ∃βA(α, β) for some stable A(α, β) (which belongs to the classical analytical hierarchy).The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems, is maximal with respect to classical Kleene function realizability, which establishes its consistency. The usual disjunction and (recursive) existence properties ensure that preserves the constructive sense of “or” and “there exists.”

This paper introduces, as an alternative to the (absolutely) lawless sequences of Kreisel and Troelstra, a notion of choice sequence lawless with respect to a given class of lawlike sequences. For countable , the class of -lawless sequences is comeager in the sense of Baire. If a particular well-ordered class of sequences, generated by iterating definability over the continuum, is countable then the -lawless sequences satisfy the axiom of open data and the continuity principle for functions from lawless to lawlike sequences, but fail to satisfy Troelstra's extension principle. Classical reasoning is used.

In [12] we defined an extensional notion of relative lawlessness and gave a classical model for a theory of lawlike, arbitrary choice, and lawless sequences. Here we introduce a corresponding intuitionistic theory and give a realizability interpretation for it. Like the earlier classical model, this realizability model depends on the (classically consistent) set theoretic assumption that a particular ∆ 2 1 well ordered subclass of Baire space is countable.