Notions of Cauchyness and Metastability

We show that several weakenings of the Cauchy condition are all equivalent under the assumption of countable choice, and investigate to what extent choice is necessary. We also show that the syntactically reminiscent notion of metastability allows similar variations, but in terms of its computational content is an empty notion. §1. Almost Cauchyness. Apart from the last section, we work in Bishop style constructive mathematics [4]—that is mathematics using intuitionistic instead of classical logic and some appropriate set-theoretic or type theoretic foundation [1]. Unlike Bishop, however, we do not freely use the axiom of countable/dependent choice, but explicitly state every such use. In [3] a weakened form of the usual Cauchy condition is considered. There a sequence (xn)n>1 in a metric space (X, d) is called almost Cauchy, if for any strictly increasing f, g : N→ N d(xf(n), xg(n))→ 0 as n → ∞. (This property will be named C2 below). Unsurprisingly, and as indicated by its name, every Cauchy sequence is almost Cauchy. In the same paper mentioned above it is also shown that Ishihara’s principle BD-N suffices to show the converse: that every almost Cauchy sequence is Cauchy. Thus the two conditions are equivalent not only in classical mathematics (CLASS), but also in Brouwer’s intuitionism (INT) and Russian recursive mathematics á la Markov (RUSS) as in all these models BD-N holds. In fact, it was only recently that it has been shown that there are models in which this principle fails [7, 10]. In this paper we will link the notion of almost Cauchyness to various other weakenings proposed by Fred Richman and investigate similarities and differences to the notion of metastability which was proposed by Terence Tao. Without further ado we will start the mathematical part of the paper with the following convention: For two natural numbers n,m the interval [n,m] will denote all natural numbers between n and m; notice that this notation does not necessitate n 6 m. Proposition 1. Consider the following conditions for a sequence (xn)n>1 in a metric space (X, d), where each condition should be read as prefaced by 1As BISH is not formalised in the same spirit as normal, everyday, mathematics is formalised, we use the phrase “model of” here somewhat loosely. Of course there are strict formalisations of BISH and the structures falsifying BD-N are models of such formalisations.