Representations of reals in reverse mathematics. (English) Zbl 1138.03012

Hirst, Jeffry L.

Bull. Pol. Acad. Sci., Math. 55, No. 4, 303-316 (2007).

The author compares four representations of reals as rapidly converging Cauchy sequences, decimal expansions, Dedekind cuts, and open Dedekind cuts. While the statement that a real in any of the above representations can be converted to any other is provable in \(\text{RCA}_0\), it is not so for conversions of sequences. For example, the statement “If \(\langle \lambda_i\rangle_{i\in \omega}\) is a sequence of Dedekind cuts, then there is a sequence \(\langle \sigma_i\rangle_{i\in \omega}\) of open cuts such that, for suitably defined equality of reals, for each \(i\), \(\lambda_i=\sigma_i\),” is equivalent over \(\text{RCA}_0\) to \(\text{ACA}_0\) (showing that the real by real conversion cannot be defined uniformly). For each pair of the representations, Hirst calibrates the proof-theoretic strength of the sequential equivalence, and in each case it turns out to be one of \(\text{RCA}_0\), \(\text{WKL}_0\), or \(\text{ACA}_0\). There are also some interesting related results. For example, for any of the four representations, \(\text{ACA}_0\) is equivalent over \(\text{RCA}_0\) to the statement: If \(\langle \tau_i\rangle_{i\in \omega}\) is a sequence of reals in the specified representation, then the set \(\{i\in\omega: \tau_i \text{ is rational}\}\) exists. There are also reverse mathematics analogs of two theorems of A. Mostowski from [“On computable sequences”, Fundam. Math. 44, 37–51 (1957; Zbl 0079.24702)], concerning change of basis in base representations of sequences of reals.

Reviewer: Roman Kossak (New York)

MSC:

03B30	Foundations of classical theories (including reverse mathematics)
03F35	Second- and higher-order arithmetic and fragments
03F60	Constructive and recursive analysis

Keywords:

real analysis; Cauchy sequence; Dedekind cut; decimal expansion; reverse mathematics; WKL; ACA

Citations:

Zbl 0079.24702