| Peer-Reviewed

Strong Reflection Principles and Large Cardinal Axioms

Received: 18 May 2013     Published: 10 June 2013
Views:       Downloads:
Abstract

In this article an possible generalization of the Löb’s theorem is considered. We proved so-called uniform strong reflection principle corresponding to formal theories which has ω-models.Main result is: let κ be an inaccessible cardinal and H_κ is a set of all sets having hereditary size less than κ,then:"¬Con" ("ZFC+" ("V=" "H" _"κ" ) )

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 3)
DOI 10.11648/j.pamj.20130203.12
Page(s) 119-127
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2013. Published by Science Publishing Group

Keywords

Löb'stheorem, Second Gödelincompleteness Theorem, Consistency, Formal System, Uniform Reflection Principles, Ω-Model Of ZFC, Standard Model Of ZFC, Inaccessible Cardinal, Weakly Compact Cardinal

References
[1] M. H. Löb, "Solution of a Problem of Leon Henkin," The Journal of Symbolic Logic, Vol. 20, No. 2, 1955, pp. 115-118.
[2] C. Smorynski, "Handbook of Mathematical Logic," Edited by J. Barwise, North-Holland Publishing Company, 1977, p.1151.
[3] T. Drucker, "Perspectives on the History of Mathematical Logic," Boston, Birkhauser, 2008, p.191.
[4] A. Marcja, C. Toffalori, "A Guide to Classical and Modern Model Theory," Springer, 2003. 371 p. Series: Trends in Logic, Vol. 19.
[5] F. W. Lawvere, C. Maurer, G. C. Wraith," Model Theory and Topoi." ISBN: 978-3-540-07164-8.
[6] D. Marker, "Model Theory: an Introduction," Graduate Texts in Mathematics, Vol. 217, Springer 2002.
[7] J.Foukzon,"GeneralizedLöb’s Theorem. Strong Reflection Principles and Large Cardinal Axioms" http://arxiv.org/abs/1301.5340
[8] J. Foukzon, "An Possible Generalization of the Löb’s Theorem," AMS Sectional Meeting AMS Special Session. Spring Western Sectional Meeting University of Colorado Boulder, Boulder, CO 13-14 April 2013. Meeting # 1089. http://www.ams.org/amsmtgs/2210_abstracts/1089-03-60.pdf
[9] P.Lindstrom, "First order predicate logic and generalized quantifiers",Theoria, Vol. 32, No.3. pp. 186-195, December 1966.
[10] F.R. Drake," Set Theory: An Introduction to Large Car-dinal",Studies in Logic and the Foundations of Mathe-matics,V.76.(1974).ISBN 0-444-105-35-2
[11] A.Kanamori, "The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings", (2003) Spriger ISBN 3-540-00384-3
[12] A. Brovykin, "On order-types of models of arith-metic". Ph.D. thesis pp.109, University of Bir-mingham 2000.
[13] W.V.O.,Quine, "New Foundation for Mathematical Logic,"American Mathematical Monthly,vol.44, {1937) pp.70-80.
[14] R.B.Jensen,"On the consistency of a slight (?) mo-Dification of Quine’s‘New Foundation,’ "Synthese, vol.19 (1969),250-263.
[15] M.R. Holmes,"The axiom of anti-foundation in Jensen’s ‘New Foundations with ur-elements,’" Bulletin de la SocieteMathematique de la Belgique, Series B
[16] M.R. Holmes,"Elementary Set Theory with a Uni-Versal Set,"vol.10 of Cahiers du Centre de Logi- que, UniversiteCatholiquede Louvain Lovain-la-Neuve, Belgium,1998.
[17] M.R. Holmes, "The Usual Model Construction for NFU Preserves Information," Notre Dame J. Formal Logic, Volume 53, Number 4 (2012), 571-580.
[18] A. Enayat, "Automorphisms, MahloCardinals,and NFU," AMS Special Session. Nonstandard mod- els of Arithmetic and Set Theory. January 15-16 2003,Baltimore, Maryland.
[19] R.M.Solovay,R.D.Arthan,J.Harrison,"The cosis-tency strengths of NFUB," Front for the arXiv, math/9707207 E.Mendelson, "Introduction to Mathematical Logic" ISBN-10:0412808307,ISBN-13:978-0412808302
Cite This Article
  • APA Style

    J. Foukzon, E. R Men’kova. (2013). Strong Reflection Principles and Large Cardinal Axioms. Pure and Applied Mathematics Journal, 2(3), 119-127. https://doi.org/10.11648/j.pamj.20130203.12

    Copy | Download

    ACS Style

    J. Foukzon; E. R Men’kova. Strong Reflection Principles and Large Cardinal Axioms. Pure Appl. Math. J. 2013, 2(3), 119-127. doi: 10.11648/j.pamj.20130203.12

    Copy | Download

    AMA Style

    J. Foukzon, E. R Men’kova. Strong Reflection Principles and Large Cardinal Axioms. Pure Appl Math J. 2013;2(3):119-127. doi: 10.11648/j.pamj.20130203.12

    Copy | Download

  • @article{10.11648/j.pamj.20130203.12,
      author = {J. Foukzon and E. R Men’kova},
      title = {Strong Reflection Principles and Large Cardinal Axioms},
      journal = {Pure and Applied Mathematics Journal},
      volume = {2},
      number = {3},
      pages = {119-127},
      doi = {10.11648/j.pamj.20130203.12},
      url = {https://doi.org/10.11648/j.pamj.20130203.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130203.12},
      abstract = {In this article an possible generalization of the Löb’s theorem is considered. We proved so-called uniform strong reflection principle corresponding to formal theories which has ω-models.Main result is: let κ be an inaccessible cardinal and H_κ is a set of all sets having hereditary size less than κ,then:"¬Con" ("ZFC+" ("V=" "H" _"κ"  ) )},
     year = {2013}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Strong Reflection Principles and Large Cardinal Axioms
    AU  - J. Foukzon
    AU  - E. R Men’kova
    Y1  - 2013/06/10
    PY  - 2013
    N1  - https://doi.org/10.11648/j.pamj.20130203.12
    DO  - 10.11648/j.pamj.20130203.12
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 119
    EP  - 127
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20130203.12
    AB  - In this article an possible generalization of the Löb’s theorem is considered. We proved so-called uniform strong reflection principle corresponding to formal theories which has ω-models.Main result is: let κ be an inaccessible cardinal and H_κ is a set of all sets having hereditary size less than κ,then:"¬Con" ("ZFC+" ("V=" "H" _"κ"  ) )
    VL  - 2
    IS  - 3
    ER  - 

    Copy | Download

Author Information
  • Israel Institute of Technology, Haifa, Israel

  • LomonosovMoscowStateUniversity, Moscow, Russia

  • Sections