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].