An intuitionistic theory of lawlike, choice and lawless sequences

Abstract

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.

More about relatively lawless sequences

“…D L S + V KL1 is inconsistent; see note at end. By [7] the systems BDLS" and IDLS" obtained by replacing the "3a" in V DLS1 by "-.Va-i" and the "3/?" in Y DLS2 by "-.V/?-."…”

“…7 The collection DLS of sequences we call "lawless relative to D" is defined by a predictability condition which entails natural closure and density properties. This definition is simpler than, but equivalent to, the one given in [6], [7]. MAIN DEFINITION. A predictor is any function n which maps sequence numbers to sequence numbers.…”

“…This y' is a lawlike permutation with the desired property. By Lemma 5 (essentially the technical lemma of [7]) with the constructive proofs given in [6]. [7].…”

“…The last axiom schema for BDLS, strengthening the theory called by the same acronym in [7]. is the principle of closed data:…”

“…As in [7] the intuitionistic theory IDLS comes form BDLS by adjoining Kleene's principle for functions (Troelstra's Generalized Continuity Principle):…”