Archive for Mathematical Logic

, Volume 47, Issue 2, pp 143–157 | Cite as

A hierarchy of hereditarily finite sets

  • Laurence Kirby


This article defines a hierarchy on the hereditarily finite sets which reflects the way sets are built up from the empty set by repeated adjunction, the addition to an already existing set of a single new element drawn from the already existing sets. The structure of the lowest levels of this hierarchy is examined, and some results are obtained about the cardinalities of levels of the hierarchy.


Finite set theory Adjunction Adduction Hierarchy 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Givant S., Tarski A.: Peano arithmetic and the Zermelo-like theory of sets with finite ranks. Not. Am. Math. Soc. 77T-E51, A-437 (1977)Google Scholar
  2. 2.
    Kirby L.: Addition and multiplication of sets. Math. Log. Q. 53(1), 52–65 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kirby, L.: Finitary set theory. Notre Dame J. Formal Log. (to appear)Google Scholar
  4. 4.
    Mathias A.R.D.: Slim models of Zermelo set theory. J. Symb. Log. 66(2), 487–496 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Previale F.: Induction and foundation in the theory of the hereditarily finite sets. Arch. Math. Log. 33, 213–241 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Shepherdson J.C.: Inner models for set theory—Part I. J. Symb. Log. 16(3), 161–190 (1951)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Tait W.W.: Finitism. J. Philos. 78(9), 524–546 (1981)CrossRefGoogle Scholar
  8. 8.
    Tarski A.: Sur les ensembles finis. Fundam. Math. 6, 45–95 (1924)Google Scholar
  9. 9.
    Wolfram S.: A New Kind of Science. Wolfram Media, Champaign (2002)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics, Baruch CollegeCity University of New YorkNew YorkUSA

Personalised recommendations