Absolutely non-computable predicates and functions in analysis

In the representation approach (TTE) to computable analysis, the representations of an algebraic or topological structure for which the basic predicates and functions become computable are of particular interest. There are, however, many predicates (like equality of real numbers) and functions that are absolutely non-computable, that is, not computable for any representation. Many of these results can be deduced from a simple lemma. In this article we prove this lemma for multi-representations and apply it to a number of examples. As applications, we show that various predicates and functions on computable measure spaces are absolutely non-computable. Since all the arguments are topological, we prove that the predicates are not relatively open and the functions are not relatively continuous for any multi-representation.

More on entangled orders

This paper grew as a continuation of [Sh462] but in the present form it can serve as a motivation for it as well. We deal with the same notions, all defined in 1.1, and use just one simple lemma from there whose statement and proof we repeat as 2.1. Originally entangledness was introduced, in [BoSh210] for example, in order to get narrow boolean algebras and examples of the nonmultiplicativity of c.c-ness. These applications became marginal when other methods were found and successfully applied (especially Todorčevic walks) but after the pcf constructions which made their début in [Sh-g] and were continued in [Sh462] it seems that this notion gained independence.Generally we aim at characterizing the existence of strong and weak entangled orders in cardinal arithmetic terms. In [Sh462, §6] necessary conditions were shown for strong entangledness which in a previous version was erroneously proved to be equivalent to plain entangledness. In §1 we give a forcing counterexample to this equivalence and in §2 we get those results for entangledness (certainly the most interesting case). A new construction of an entangled order ends this section. In §3 we get weaker results for positively entangledness, especially when supplemented with the existence of a separating point (Definition 2.2). An antipodal case is defined in 3.10 and completely characterized in 3.11. Lastly we outline in 3.12 a forcing example showing that these two subcases of positive entangledness comprise no dichotomy. The work was done during the fall of 1994 and the winter of 1995. The second author proved Theorems 1.2, 2.14, the result that is mentioned in Remark 2.11 and what appears in this version as Theorem 2.10(a) with the further assumption den (I)θ < μ. The first author is responsible for waving off this assumption (actually by showing that it holds in the general case), for Theorems 2.12 and 2.13 in Section 2 and for the work which is presented in Section 3.