Being able to state the principles which lie deepest in the
foundations of mathematics by sentences in three variables is
crucially important for a satisfactory equational rendering of set
theories along the lines proposed by Alfred Tarski and Steven
Givant in their monograph of 1987.
¶
The main achievement of this paper is the proof that the ‘kernel’
set theory whose postulates are extensionality,
(E), and single-element adjunction and
removal, (W) and (L), cannot be
axiomatized by means of three-variable sentences. This highlights
a sharp edge to be crossed in order to attain an ‘algebraization’
of Set Theory. Indeed, one easily shows that the theory which
results from the said kernel by addition of the null set axiom,
(N), is in its entirety expressible in three variables.