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The axiom of elementary sets on the edge of Peircean expressibility

Published online by Cambridge University Press:  12 March 2014

Andrea Formisano
Affiliation:
Dipartimento di Informatica, UniversitÀ di Laquila, Via Vetoio–Loc, Coppito, 67010 l'Aquila, ItalyE-mail:, formisano@di.univaq.it
Eugenio G. Omodeo
Affiliation:
Dipartimento di Matematica e Informatica, UniversitÀ di Trieste, Via Valerio, 12/B, 34127 Trieste, ItalyE-mail:, eomodeo@univ.trieste.it
Alberto Policriti
Affiliation:
Dipartimento di Matematica e Informatica, UniversitÀ di Udine, Via Delle Scienze 206-Loc. Rizzi, 33100 Udine, ItalyE-mail:, policrit@dimi.uniud.it

Abstract

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2005

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References

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