Journal of Symbolic Logic

The axiom of elementary sets on the edge of Peircean expressibility

Andrea Formisano, Eugenio G. Omodeo, and Alberto Policriti

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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.

Article information

Source
J. Symbolic Logic, Volume 70, Issue 3 (2005), 953-968.

Dates
First available in Project Euclid: 22 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1122038922

Digital Object Identifier
doi:10.2178/jsl/1122038922

Mathematical Reviews number (MathSciNet)
MR2155274

Zentralblatt MATH identifier
1100.03042

Keywords
Weak set theories n-variable expressibility pebble games

Citation

Formisano, Andrea; Omodeo, Eugenio G.; Policriti, Alberto. The axiom of elementary sets on the edge of Peircean expressibility. J. Symbolic Logic 70 (2005), no. 3, 953--968. doi:10.2178/jsl/1122038922. https://projecteuclid.org/euclid.jsl/1122038922


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