On the ‘definability of definable’ problem of Alfred Tarski, Part II

Abstract

Alfred Tarski [J. Symbolic Logic 13 (1948), pp. 107–111] defined D p m \mathbf {D}_{pm} to be the set of all sets of type p p , type-theoretically definable by parameterfree formulas of type ≤ m {\le m} , and asked whether it is true that D 1 m ∈ D 2 m \mathbf {D}_{1m}\in \mathbf {D}_{2m} for m ≥ 1 m\ge 1 . Tarski noted that the negative solution is consistent because the axiom of constructibility V = L \mathbf {V}=\mathbf {L} implies D 1 m ∉ D 2 m \mathbf {D}_{1m}\notin \mathbf {D}_{2m} for all m ≥ 1 m\ge 1 , and he left the consistency of the positive solution as a major open problem. This was solved in our recent paper [Mathematics 8 (2020), pp. 1–36], where it is established that for any m ≥ 1 m\ge 1 there is a generic extension of L \mathbf {L} , the constructible universe, in which it is true that D 1 m ∈ D 2 m \mathbf {D}_{1m}\in \mathbf {D}_{2m} . In continuation of this research, we prove here that Tarski’s sentences D 1 m ∈ D 2 m \mathbf {D}_{1m}\in \mathbf {D}_{2m} are not only consistent, but also independent of each other, in the sense that for any set Y ⊆ ω ∖ { 0 } Y\subseteq \omega \smallsetminus \{0\} in L \mathbf {L} there is a generic extension of L \mathbf {L} in which it is true that D 1 m ∈ D 2 m \mathbf {D}_{1m}\in \mathbf {D}_{2m} holds for all m ∈ Y m\in Y but fails for all m ≥ 1 m\ge 1 , m ∉ Y m\notin Y . This gives a full and conclusive solution of the Tarski problem. The other main result of this paper is the consistency of D 1 ∈ D 2 \mathbf {D}_{1}\in \mathbf {D}_{2} via another generic extension of L \mathbf {L} , where D p = ⋃ m D p m \mathbf {D}_{p}=\bigcup _m\mathbf {D}_{pm} , the set of all sets of type p p , type-theoretically definable by formulas of any type. Our methods are based on almost-disjoint forcing of Jensen and Solovay [Some applications of almost disjoint sets, North-Holland, Amsterdam, 1970, pp. 84–104].