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The axiom of elementary sets on the edge of Peircean expressibility. (English) Zbl 1100.03042
Summary: 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 A. Tarski and S. Givant in their monograph [A formalization of set theory without variables. Providence, RI: American Mathematical Society (1987; Zbl 0654.03036)]. The main achievement of this paper is the proof that the ‘kernel’ set theory, whose postulates are extensionality, \(({\mathbf E})\), 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.

MSC:
03E30 Axiomatics of classical set theory and its fragments
Citations:
Zbl 0654.03036
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