Specker sequence

Article summary

in a computable Gödel numbering halts on input n after k steps; otherwise it is 3. In computability theory, a Specker sequence is a computable, monotonically increasing, bounded sequence of rational numbers whose supremum is not a computable real number. The first example of such a sequence was constructed by Ernst Specker. The existence of Specker sequences has consequences for computable analysis. The fact that such sequences exist means that the collection of all computable real numbers does not satisfy the least upper bound principle of real analysis, even when considering only computable sequences. A common way to resolve this difficulty is to consider only sequences that are accompanied by a modulus of convergence; no Specker sequence has a computable modulus of convergence. More generally, a Specker sequence is called a recursive counterexample to the least upper bound principle, i.e. a construction that shows that this theorem is false when restricted to computable reals. The least upper bound principle has also been analyzed in the program of reverse mathematics, where the exact strength of this principle has been determined. In the terminology of that program, the least upper bound principle is equivalent to ACA 0 over RCA 0 . In fact, the proof of the forward implication, i.e.…

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