Joan Rand Moschovakis

Relative Lawlessness in Intuitionistic Analysis

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 Troelstras extension principle. Classical reasoning is used.

Some axioms for constructive analysis

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CFd asserting that every decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC00 of countable choice can be decomposed into a monotone choice schema AC00m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC00. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms

Can There be no Nonrecursive Functions

{AB_{1/2,0}^{2^{\mathbb{N}}}}

More about relatively lawless sequences

. We also introduce a version WKL!! of Weak König’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FTd.

A disjunctive decomposition theorem for classical theories

Five recursively axiomatizable theories extending Kleenes intuitionistic theory FIM of numbers and numbertheoretic (choice) sequences are introduced and shown to be consistent, by a modified relative realizability interpretation which verifies that every sequence classically defined by a Π11 formula is unavoidable (cannot fail to exist) and that no sequence can fail to be classically Δ11. The analytical form of Markovs Principle fails under the interpretation. The notion of strongly inadmissible rule of inference is introduced, with examples (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)