More about relatively lawless sequences

Abstract

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 lawlessness 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.

An intuitionistic theory of lawlike, choice and lawless sequences

“…Here α • γ can be thought of as a subpermutation of α, so α is lawless if and only if every lawlike predictor is eventually correct (and hence very often wrong) on every strongly lawlike subpermutation of α. 11 Note added in proof: A simpler, but equivalent, definition of "lawless (relative to D)" appears in [13].…”

“…(c) If α is lawless, so is λy α((x, y)) for each x ∈ N. (d) If w is any sequence number and α is lawless then w * α is lawless. 13 Lemma 2. (First Density Lemma.)…”

Constructivity and Computability in Historical and Philosophical Perspective

“…In a series of papers (Moschovakis 1993(Moschovakis , 1994(Moschovakis , 1996, Joan Rand Moschovakis introduced a very convincing notion of chaotic sequence˛2 f0; 1g N . It turns out that the set of such sequences has measure zero and is disjoint from Martin-Löf random sequences.…”

Is Randomness Native to Computer Science? Ten Years After

“…Though they are unpredictable, random sequences are not lawless. An interesting notion of lawless sequence has been introduced by Joan Moschovakis, 1987-94 [23,24].…”