Analysing choice sequences

Relative lawlessness in intuitionistic analysis

“…This paper will introduce a defined class of sequences lawless relative to a given class D of lawlike sequences. Relatively lawless sequences differ fundamentally from Kreisel's (absolutely) lawless sequences and the proto-lawless sequences of Troelstra [1983], which cannot be considered as classical objects. Since relatively lawless sequences can be unequal without being independent, they do not satisfy all the axioms of LS; nevertheless, for a particular countable D obtained by transfinite iteration of definability over the continuum, the relatively lawless sequences provide a classical model for a closely related theory DLS, which therefore is compatible with classical logic, and which includes a continuity principle for functions from lawless to lawlike sequences.…”

“…In [1957] and [1965] S. C. Kleene formalized Brouwer's analysis with a strong form of the continuity principle and gave a constructive function-realizability interpretation for his theory, thus establishing its consistency. In [1968] G. Kreisel proposed a theory LS of lawlike (proceeding according to fixed law) and lawless (eventually restricted by no law) sequences, and sketched a proof that any statement involving lawless and lawlike sequence quantifiers was equivalent to some purely lawlike statement; for details and a correction to the axioms see A. S. Troelstra [1977]. A similar result, for a theory CS of lawlike and arbitrary choice sequences (a conservative extension of Kleene's theory), appeared in Kreisel and Troelstra [1970].…”

“…Each of these axiomatizations began as an attempt to elucidate Brouwer's vision of the continuum, and ended by extending it in one direction or another: Kleene's strong continuity principle, Kreisel's axiom of open data, and Troelstra's axiom of analytic data all go beyond the principles explicit in Brouwer's published work. Troelstra [1983] eloquently justifies the search for new concepts of choice sequence in the spirit of Brouwer, whose own concept of the continuum evolved radically from the geometrical "filling-in" of the rationals in his doctoral dissertation [1907] to the analytic structure of his retrospective survey [1952]. Kleene's elegant axiomatization (Kleene and Vesley [1965]), in addition to neatly separating the specifically intuitionistic content of Brouwer's analysis from the basic principles accepted also by classical mathematicians, provided a framework within which definable subclasses of the continuum (such as the general recursive functions) could readily be studied (Kleene [1968]).…”

“…In the language of LS, x, y, x t ,... vary over natural numbers, m, n, m x ,... over finite sequences of numbers, a, b, a t ,... over lawlike sequences, e, f, e l 5 ... over K, and a, P, y, a 1 ; ... over lawless sequences. Slightly modifying the notation of Troelstra [1977], and choosing his starred form of the axiom of open data, we may state the new axioms as follows. In LS3 and LS4, A must contain free no lawless variables except those explicitly shown.…”

Constructing choice sequences from lawless sequences of neighbourhood functions

“…(See the remarks in Troelstra(1983Troelstra( ), pp. 2456 Thus, the constructions of these three CSmodels are motivated rather different ly, and the relation between these models needs closer investigation.…”