Journal of Symbolic Logic 65 (2):857-884 (2000)

Abstract
Here we investigate the classes RCA $^\uparrow_\alpha$ of representable directed cylindric algebras of dimension α 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 α ≥ 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
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DOI 10.2307/2586575
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Non-Well-Founded Sets.J. L. Bell - 1989 - Journal of Symbolic Logic 54 (3):1111-1112.

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Three Interpolation Theorems for Typeless Logics.T. Sayed Ahmed - 2012 - Logic Journal of the IGPL 20 (6):1001-1037.
Omitting Types for Algebraizable Extensions of First Order Logic.Tarek Sayed Ahmed - 2005 - Journal of Applied Non-Classical Logics 15 (4):465-489.

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