Skip to main content
SpringerLink
Log in

Localizing the axioms

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

An Erratum to this article was published on 17 November 2010

Abstract

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All \({\Pi_2}\) consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ0-Collection and minus \({\in}\) -induction scheme. ZFC+ “there is an inaccessible cardinal” proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and \({\Pi_1^1}\) -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form \({Loc({\rm ZFC}+\phi)}\) are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barwise J.: Admisssible Sets and Structures, Perspectives in Mathematical Logic. Springer, Berlin (1975)

    Google Scholar 

  2. Chang C.C., Keisler H.J.: Model Theory. North Holland, Amsterdam (1973)

    MATH  Google Scholar 

  3. Drake F.R.: Set Theory, An Introduction to Large Cardinals. North Holland, Amsterdam (1974)

    MATH  Google Scholar 

  4. Enayat, A.: Automorphisms and Mahlo cardinals, Non Standard Models of Arithmetic and Set Theory. In: Enayat, A., Kossak, R. (eds.) Contemp. Math. 361, 37–59 (2004)

  5. Feferman S.: Transfinite recursive progressions of axiomatic theories. J. Symb. Log. 27, 259–316 (1962)

    Article  MathSciNet  Google Scholar 

  6. Jech T.: Set Theory, the Third Millenium Edition. Springer, Berlin (2003)

    Google Scholar 

  7. Kanamori A.: The Higher Infinite Perspectives in Mathematical Logic. Springer, Berlin (1997)

    Google Scholar 

  8. Montague R., Vaught R.L.: Natural models of set theories. Fund. Math. XLVII, 219–242 (1959)

    MathSciNet  Google Scholar 

  9. Skolem T.: Some remarks on axiomatized set theory. In: Heijenoort, Jean van (eds) From Frege to Gödel A Source Book in Mathematical Logic 1879–1931, pp. 290–301. Third Printing, Harvard U.P. (1976)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Athanassios Tzouvaras

Authors
  1. Athanassios Tzouvaras
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Athanassios Tzouvaras.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s00153-010-0217-4

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tzouvaras, A. Localizing the axioms. Arch. Math. Logic 49, 571–601 (2010). https://doi.org/10.1007/s00153-010-0188-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-010-0188-5

Keywords

Mathematical Subject Classification (2000)

Search

Navigation