Dialectical categories, cardinalities of the continuum and combinatorics of ideals

Abstract

Andreas Blass has frequently pointed out that inequalities between cardinal invariants of the continuum are usually proved via morphisms of some versions of (dual) Dialectica Categories-which are certain categories introduced by the second author as categorical models of linear logic. In this paper, we discuss the reasons why Dialectica Categories can be successfully applied to prove such inequalities. The main goal of this ongoing research is to circumscribe the effectivity of the described method and to discover why it works as well as it does. Combinatorics of ideals and the notions of unboundedness and domination in pre-orders are presented as study cases which serve as evidence in favour of some conjectural principles. To finish, a number of questions and problems are posed.

Reductions between certain incidence problems and the continuum hypothesis

“…The application of PV to Set Theory which was extensively studied by Blass in the 90's (see e.g. [3], [4]), and, more recently, by the author ( [19], [6], [20]), is the so-called method of morphisms in the proof of inequalities between cardinal invariants of the continuum -which was once declared by Blass as being an empirical fact. 3 The ideal I has to satisfy |I| 2 ℵ 0 in order to certain objects (which will be presently described) be, formally, objects of PV; notice that this is the case for the ideal of all countable subsets of R. However, it is worthwhile mentioning that for the ideals M of all meager subsets of R and L of all null subsets of R (which both have size 2 2 ℵ 0 ), the corresponding objects are considered as being objects of PV within the literature, and this is justified by the fact that each of those two ideals have a basis of Borel sets; recall that there are exactly 2 ℵ 0 Borel subsets of R.…”

“…Several connections between such category and Set Theory have been extensively studied by Andreas Blass in the 90's (see e.g. [3], [4]), and, more recently, have also been investigated by the author ( [19], [20]).…”