V. Lifschitz defined in 1979 a variant of realizability which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis. In this paper we describe an extension of intuitionistic arithmetic in which the soundness of Lifschitz' realizability can be proved, and we give an axiomatic characterization of the Lifschitz-realizable formulas relative to this extension. By a "q-variant" we obtain a new derived rule. We also show how to extend Lifschitz' realizability to second-order arithmetic. Finally we describe (...) an analogous development for elementary analysis, with partial continuous application replacing partial recursive application. (shrink)

In the author's Relative lawlessness in intuitionistic analysis [this JOURNAL. vol. 52 (1987). pp. 68-88] and An intuitionistic theory of lawlike, choice and lawless sequences [Logic Colloquium '90. Springer-Verlag. Berlin. 1993. pp. 191-209] a notion of lawless ness relative to a countable information base was developed for classical and intuitionistic analysis. Here we simplify the predictability property characterizing relatively lawless sequences and derive it from the new axiom of closed data (classically equivalent to open data) together with a natural principle (...) of invariance under finite translation. We characterize relative lawlessness in terms of a notion of forcing. Finally, we study relative lawlessness on an arbitrary fan and show that the collection of lawless binary sequences (which is comeager in the sense of Baire) has probability measure zero. The reasoning is predominantly constructive. (shrink)

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 D of lawlike sequences. For countable D, the class of D-lawless sequences is comeager in the sense of Baire. If a particular well-ordered class F of sequences, generated by iterating definability over the continuum, is countable then the F-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. (shrink)

When Kleene extended his recursive realizability interpretation from intuitionistic arithmetic to analysis, he was forced to use more than recursive functions to interpret sequences and conditional constructions. In fact, he used what classically appears to be the full continuum. We describe here a generalization to higher type of Kleene's realizability, one case of which, -realizability, uses general recursive functions throughout, both to realize theorems and to interpret choice sequences. -realizability validates a version of the bar theorem and the usual continuity (...) principles, while also providing naturally, as Kleene's 1965 realizability does not, for versions of lawless sequence axioms, as well as of Church's Thesis. (shrink)