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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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\textit{A. Formisano} et al., J. Symb. Log. 70, No. 3, 953--968 (2005; Zbl 1100.03042)

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### References:

[1] | Finite model theory (1999) |

[2] | Finite models and finitely many variables 46 (1999) |

[3] | On Moschovakis closure ordinals 42 pp 292– (1977) · Zbl 0367.02021 |

[4] | Handbook of formal languages III pp 389– (1997) |

[5] | A formalization of Set Theory without variables 41 (1987) · Zbl 0654.03036 |

[6] | Some metalogical results concerning the calculus of relations 18 pp 188– (1953) |

[7] | DOI: 10.1016/j.tcs.2004.03.028 · Zbl 1058.03028 |

[8] | Proceedings, 11th annual IEEE symposium on logic in computer science pp 348– (1996) |

[9] | DOI: 10.1016/0022-0000(82)90011-3 · Zbl 0503.68032 |

[10] | Bulletin of the European Association for Theoretical Computer Science 51 pp 111– (1993) |

[11] | From Frege to GÖdel– A source book in mathematical logic, 1879–1931 (1977) |

[12] | Journal of Symbolic Computation 29 (2000) |

[13] | DOI: 10.1016/0890-5401(89)90055-2 · Zbl 0711.03004 |

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