Skip to main content

Reflection in Set Theory the Bernays-Levy Axiom System

  • Chapter
  • First Online:
Philosophy of Mathematics Today

Part of the book series: Episteme ((EPIS,volume 22))

  • 489 Accesses

Abstract

It is a classical saying that mathematics is about numbers and space. Of course, in the last three centuries, the theory of movement and more important statistics also entered the scene. But still two other subjects of independent character became treatable by mathematical methods, namely infinity in the proper sense (Cantor) and a theory of coding (Gödel). These two themes are clearly related to philosophy in general and especially to epistemology (“Erkenntnistheorie”). This is evident in the case of any theory of coding. The relation of a code to what it is supposed to code — and many related questions — became treatable inside mathematics. The phenomena, results and observations may and will lead to a better and deeper understanding of the role and the merits or weaknesses of coding in the general sense of epistemology. Here I will not enter into this fascinating theme.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Beeson, M. J. [1980] Foundations of Constructive Mathematics, Berlin: Springer Verlag.

    MATH  Google Scholar 

  • Bernays, P. [1976] On the problem of schemata of infinity in axiomatic set theory, in: G. H. Müller [1976], pp. 121–172, (see also in: Essays on the Foundations of Mathematics, (Fraenkel anniversary vol.), [1961], pp. 3–49, Jerusalem: Magnes Press).

    Google Scholar 

  • Bernays, P. [1937] A system of axiomatic set theory, Journal of Symbolic Logic, 65–77, (and in G. H. Müller [1976]).

    Google Scholar 

  • Dummett, M. [1977] Elements of Intuitionism, Oxford Log. Guides, Oxford: Clarendon Press.

    MATH  Google Scholar 

  • Gloede, K. [1976] Reflection principles and indescribability, in: G. H. Müller, [1976], pp. 277–323.

    MATH  Google Scholar 

  • Kanamori, A., Magidor, M. [1978] The evolution of large cardinal axioms in set theory, in: G. H. Müller, D. S. Scott [1978], pp. 99–275.

    Google Scholar 

  • Levy, A. [1958] Contributions to the Metamathematics of Set Theory, [Ph.D. thesis], Jerusalem, [in Hebrew, with ext. English abstract).

    Google Scholar 

  • Levy, A. [1971] The sizes of the indescribable cardinals, in: D. S. Scott, ed., Proceedings of the Symposium on Pure Mathematics, vol. 13, part I, pp. 205–218.

    Google Scholar 

  • Los, J. [1955] Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres, in: Mathematical Interpretation of Formal Systems, pp. 98–113, Amsterdam: North Holland Publ. Co..

    Chapter  Google Scholar 

  • Mac Lane, S. [1986] Mathematics Form and Function, Berlin: Springer Verlag.

    Book  Google Scholar 

  • Maddy, P. [1988] Believing the axioms I and II, Journal of Symbolic Logc, 53, 481–511 and 736–764.

    Article  MathSciNet  Google Scholar 

  • Mahlo, P. [1911] Über lineare transfinite Mengen, Berichte der königlich sächsischen Gesellschaft der Wissenschaften, Leipzig, Mathematische Klasse, 61,187–225.

    MATH  Google Scholar 

  • Moore, G. H. [1982] Zermelo’s axioms of choice, Berlin: Springer Verlag.

    Book  Google Scholar 

  • Müller, G. H., (ed.) [1976] Sets and classes, Amsterdam: North Holland Publ. Co..

    Google Scholar 

  • Müller, G. H. [1980] Hierarchies and closure properties, (extended abstract), in: Proceeding of the Symposium on the Foundations of Mathematics, Gora, Hakone (Japan), pp. 35–37.

    Google Scholar 

  • Müller, G. H. [1981] Framing mathematics, Epistemologia, 4,253–285.

    Google Scholar 

  • Müller, G. H. [1988] Shadows of infinity, Epistemologia, 11,197–208.

    Google Scholar 

  • Müller, G. H., Scott, D. S., (eds.) [1978] Higher Set Theory, Lecture Notes on Mathematics, No. 669, Berlin: Springer Verlag.

    Google Scholar 

  • Reinhardt, W. M. [1974] Remarks on reflection principles, large cardinals, and elementary embeddings, in: T. J. Jech, (ed.), Proceedings of the Symposium on Pure Mathematics, vol. 13, part II, pp. 189–205.

    Google Scholar 

  • Scott, D. S. [1961] Measurable cardinals and constructible sets, Bulletin, Académie Polonaise des Sciences, Ser. Sci. Math., 7,145–149.

    MATH  Google Scholar 

  • Solovay, R. M., Reinhardt, R.M. and Kanamori, A. [1978] Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, 13,73–116.

    Article  MathSciNet  Google Scholar 

  • Specker, E. [1976] (Remark in P. Bernays [1976] pp. 137–138, and [1961] pp. 22–23).

    Google Scholar 

  • Takeuti, G. [1969] The universe of set theory, in: Foundations of Mathematics, (Gödel anniversary vol.), Berlin: Springer Verlag, pp. 74–128.

    Chapter  Google Scholar 

  • Wittgenstein, L. [1922] Tractatus Logico-Philosophicus, London: Kegan Paul, Trench, Trubner & Co..

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Müller, G.H. (1997). Reflection in Set Theory the Bernays-Levy Axiom System. In: Agazzi, E., Darvas, G. (eds) Philosophy of Mathematics Today. Episteme, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5690-5_9

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-5690-5_9

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6400-2

  • Online ISBN: 978-94-011-5690-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics