Here we investigate the classes RCA$^\uparrow_\alpha$ of representable directed cylindric algebras of dimension $\alpha$ introduced by Nemeti[12]. RCA$^\uparrow_\alpha$ can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, "purely cylindric algebraic" proof for the following theorems of Nemeti: (i) RCA$^\uparrow_\alpha$ is a finitely axiomatizable variety whenever $\alpha \geq 3$ is finite and (ii) one can obtain a strong representation theorem for RCA$^\uparrow_\alpha$ if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories.