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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 ,
Eugenio G. Omodeo  and
Alberto Policriti
Andrea Formisano
Affiliation:
Dipartimento di Informatica, UniversitÀ di Laquila, Via Vetoio–Loc, Coppito, 67010 l'Aquila, ItalyE-mail:, [email protected]
Eugenio G. Omodeo
Affiliation:
Dipartimento di Matematica e Informatica, UniversitÀ di Trieste, Via Valerio, 12/B, 34127 Trieste, ItalyE-mail:, [email protected]
Alberto Policriti
Affiliation:
Dipartimento di Matematica e Informatica, UniversitÀ di Udine, Via Delle Scienze 206-Loc. Rizzi, 33100 Udine, ItalyE-mail:, [email protected]
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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.

Keywords

Weak set theoriesn-variable expressibilitypebble games
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 953 - 968
Copyright
Copyright © Association for Symbolic Logic 2005

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