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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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

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

First available in Project Euclid: 22 July 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Weak set theories n-variable expressibility pebble games


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.

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  • J. Barwise On Moschovakis closure ordinals, Journal of Symbolic Logic, vol. 42 (1977), pp. 292--296.
  • A. Dawar Finite models and finitely many variables, Banach Center Publications, vol. 46, Institute of Mathematics, Polish Academy of Sciences,1999.
  • H.-D. Ebbinghaus and J. Flum Finite model theory, Perspectives in Mathematical Logic, Springer,1999, Second revised and enlarged edition.
  • A. Formisano, E. G. Omodeo, and P. Policriti Three-variable statements of set-pairing, Theoretical Computer Science, vol. 322 (2004), no. 1, pp. 147--173.
  • A. Formisano, E. G. Omodeo, and M. Temperini Goals and benchmarks for automated map reasoning, Journal of Symbolic Computation, vol. 29 (2000), no. 2, Special issue. (M.-P. Bonacina and U. Furbach, editors).
  • J. van Heijenoort (editor) From Frege to Gödel --- A source book in mathematical logic, 1879--1931, $3^rd$ printing ed., Source books in the history of the sciences, Harvard University Press,1977.
  • I. Hodkinson Finite variable logics, Bulletin of the European Association for Theoretical Computer Science, vol. 51 (1993), pp. 111--140, Columns: Logic in Computer Science.
  • N. Immerman Upper and lower bounds for first order expressibility, Journal of Computer and System Sciences, vol. 25 (1982), no. 1, pp. 76--98.
  • N. Immerman and D. Kozen Definability with bounded number of bound variables, Information and Computation, vol. 83 (1989), no. 2, pp. 121--139.
  • P. G. Kolaitis and M. Y. Vardi On the expressive power of variable-confined logics, Proceedings, 11$^\mathrmth$ annual IEEE symposium on logic in computer science (New Brunswick, New Jersey), IEEE Computer Society Press,1996, pp. 348--359.
  • M. K. Kwatinetz Problems of expressibility in finite languages, Ph.D. thesis, University of California, Berkeley,1981.
  • A. Tarski Some metalogical results concerning the calculus of relations, Journal of Symbolic Logic, vol. 18 (1953), pp. 188--189.
  • A. Tarski and S. Givant A formalization of Set Theory without variables, Colloquium Publications, vol. 41, American Mathematical Society,1987.
  • W. Thomas Languages, automata and logic, Handbook of formal languages, vol. III (G. Rozenberg and Salomaa A., editors), Springer,1997, pp. 389--455.
  • E. Zermelo Untersuchungen über die Grundlagen der Mengenlehre I, In Heijenoort [?], (English translation), pp. 199--215.