  Cylindric algebras, or concept algebras in another name, form an interface
between algebra, geometry and logic; they were invented by Alfred Tarski around
1947. We prove that there are 2 to the alpha many varieties of geometric (i.e.,
representable) alpha-dimensional cylindric algebras, this means that 2 to the
alpha properties of definable relations of (possibly infinitary) models of
first-order logic theories can be expressed by formula schemes using alpha
variables, where alpha is infinite. This solves Problem 4.2 in the 1985
Henkin-Monk-Tarski monograph. For solving this problem, we had to devise a new
kind of construction, which we then use to solve Problem 2.13 of the 1971
Henkin-Monk-Tarski monograph which concerns the structural description of
geometric cylindric algebras. There are fewer varieties generated by locally
finite dimensional cylindric algebras, and we get a characterization of these
among all the 2 to the alpha varieties. As a by-product, we get a simple,
natural recursive enumeration of all the equations true of geometric cylindric
algebras, and this can serve as a solution to Problem 4.1 of the 1985
Henkin-Monk-Tarski monograph. All this has logical content and implications
concerning ordinary first order logic with a countable number of variables.
