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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beeson, M. J. [1980] Foundations of Constructive Mathematics, Berlin: Springer Verlag.
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).
Bernays, P. [1937] A system of axiomatic set theory, Journal of Symbolic Logic, 65–77, (and in G. H. Müller [1976]).
Dummett, M. [1977] Elements of Intuitionism, Oxford Log. Guides, Oxford: Clarendon Press.
Gloede, K. [1976] Reflection principles and indescribability, in: G. H. Müller, [1976], pp. 277–323.
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.
Levy, A. [1958] Contributions to the Metamathematics of Set Theory, [Ph.D. thesis], Jerusalem, [in Hebrew, with ext. English abstract).
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.
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..
Mac Lane, S. [1986] Mathematics Form and Function, Berlin: Springer Verlag.
Maddy, P. [1988] Believing the axioms I and II, Journal of Symbolic Logc, 53, 481–511 and 736–764.
Mahlo, P. [1911] Über lineare transfinite Mengen, Berichte der königlich sächsischen Gesellschaft der Wissenschaften, Leipzig, Mathematische Klasse, 61,187–225.
Moore, G. H. [1982] Zermelo’s axioms of choice, Berlin: Springer Verlag.
Müller, G. H., (ed.) [1976] Sets and classes, Amsterdam: North Holland Publ. Co..
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.
Müller, G. H. [1981] Framing mathematics, Epistemologia, 4,253–285.
Müller, G. H. [1988] Shadows of infinity, Epistemologia, 11,197–208.
Müller, G. H., Scott, D. S., (eds.) [1978] Higher Set Theory, Lecture Notes on Mathematics, No. 669, Berlin: Springer Verlag.
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.
Scott, D. S. [1961] Measurable cardinals and constructible sets, Bulletin, Académie Polonaise des Sciences, Ser. Sci. Math., 7,145–149.
Solovay, R. M., Reinhardt, R.M. and Kanamori, A. [1978] Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, 13,73–116.
Specker, E. [1976] (Remark in P. Bernays [1976] pp. 137–138, and [1961] pp. 22–23).
Takeuti, G. [1969] The universe of set theory, in: Foundations of Mathematics, (Gödel anniversary vol.), Berlin: Springer Verlag, pp. 74–128.
Wittgenstein, L. [1922] Tractatus Logico-Philosophicus, London: Kegan Paul, Trench, Trubner & Co..
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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