Computable Closed Euclidean Subsets with and without Computable Points

TL;DR

This paper explores the existence of computable points in various classes of closed Euclidean subsets, revealing that non-empty co-r.e. sets without computable points must have continuum cardinality, highlighting size constraints.

Contribution

It establishes that non-empty co-r.e. closed sets without computable points must have continuum size, and investigates conditions under which computable points must exist in different classes.

Findings

Non-empty co-r.e. closed sets without computable points have continuum cardinality.

Certain size constraints are necessary for the existence of computable points.

The paper characterizes classes of computable real sets that necessarily contain computable points.

Abstract

The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskii, Tseitin, Kreisel, and Lacombe assert the existence of NON-empty co-r.e. closed sets devoid of computable points: sets which are `large' in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every non-empty co-r.e. closed real set without computable points has continuum cardinality. This leads us to investigate for various classes of computable real subsets whether they necessarily contain a (not necessarily effectively findable) computable point.