nLab Specker sequence

Definition

A Specker sequence is a bounded computable increasing infinite sequence of rational numbers with no computable supremum.

Examples

Since there is a sequence of all Turing machines, define a sequence $(S_n)_n$ as

$$S_n = \sum_{i + j = n} 2^{-i} \{i\}_j ,$$

where $\{i\}_j$ is the bit ($0$ or $1$) giving whether the $i$th Turing machine halts before $j$ steps. The theoretical limit of this sequence is

$$\sum_i 2^{-i} \{i\} ,$$

where $\{i\}$ is the bit giving whether the $i$th Turing machine halts at all, but this is uncomputable (on pain of solving the halting problem).

Many famous non-computable numbers may be expressed as limits of Specker sequences. For example, a Chaitin constant? is defined as the sum of an infinite series with non-negative computable terms (each of which is a dyadic rational), and so is the limit of a Specker sequence.

In constructive mathematics

In Russian constructivism, all real numbers are computable, so a Specker sequence has no (located) supremum, giving a counterexample to the classical least upper bound principle? ($LUP$).

In many other varieties of constructive mathematics, the computability of all real numbers can be neither proved nor refuted, but Specker sequences still provide weak counterexamples.

For a non-computable number that can be expressed as the limit of a Specker sequence, the sequence itself is a way to work with the number in constructive mathematics without assuming that it exists as a real number. (This amounts to treating it as a computable lower real number.)