Abstract
We review some fundamental questions concerning the real line of mathematical analysis, which, like the Continuum Hypothesis, are also independent of the axioms of set theory, but are of a less ‘problematic’ nature, as they can be solved by adopting the right axiomatic framework. We contend that any foundations for mathematics should be able to simply formulate such questions as well as to raise at least the theoretical hope for their resolution.
The usual procedure in set theory (as a foundation) is to add so-called strong axioms of infinity to the standard axioms of Zermelo-Fraenkel, but then the question of their justification becomes to some people vexing. We show how the adoption of a view of the universe of sets with classes, together with certain kinds of Global Reflection Principles resolves some of these issues.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
We have not exactly delineated modern indescribability properties here, which usually are defined with an extra free predicate symbol, but we only wish to give the flavour of this. See Kanamori (2003), I.6, for a fuller discussion.
References
Bernays, P. (1961). Zur Frage der Unendlichkeitsschemata in der axiomatische Mengenlehre. In Essays on the foundations of mathematics (pp. 3–49). Hebrew University of Jerusalem: Magnus Press.
Feferman, S. (2012). Is the continuum hypothesis a definite mathematical problem? In: EFI Workshop Papers, Harvard.
Friedman, H. (1970). Higher set theory and mathematical practice. Annals of Mathematical Logic, 2(3), 325–327.
Kanamori, A. (2003). The higher infinite (Springer monographs in mathematics, 2nd ed.). New York: Springer.
Koellner, P. (2009). On reflection principles. Annals of Pure and Applied Logic, 157, 206–219.
Levy, A. (1960). Axiom schemata of strong infinity in axiomatic set theory. Pacific Jouranl of Mathematics, 10, 223–238.
Lusin, N., & Sierpiński, W. (1923). Sur un ensemble non measurable B. Journal de Mathématiques, 9e serie, 2, 53–72.
Maddy, P. (1983). Proper classes. Journal for Symbolic Logic, 48(1), 113–139.
Martin, D. A. (1970). Measurable cardinals and analytic games. Fundamenta Mathematicae, 66, 287–291.
Martin, D. A. (1975). Borel determinacy. Annals of Mathematics, 102, 363–371.
Martin, D. A., & Steel, J. R. (1989). A proof of projective determinacy. Journal of the American Mathematical Society, 2, 71–125.
Moschovakis, Y. N. (2009). Descriptive set theory (Studies in logic series). Amsterdam: North-Holland.
Mycielski, J. (1964). On the axiom of determinateness. Fundamenta Mathematicae, 53, 205–224.
Mycielski, J. (1966). On the axiom of determinateness II. Fundamenta Mathematicae, 59, 203–212.
Neeman, I. (2007). Determinacy in
, chapter 22. In M. Magidor, M. Foreman, & A. Kanamori (Eds.), Handbook of set theory (vol. III). Berlin/New York: Springer.Reinhardt, W. (1974). Remarks on reflection principles, large cardinals, and elementary embeddings. In T. Jech (Ed.), Axiomatic set theory, vol. 13 part 2 of Proceedings of Symposia in Pure Mathematics (pp. 189–205). Providence: American Mathematical Society.
Roberts, S. (2017). A strong reflection principle. Review of Symbolic Logic, 10(4), 651–662.
Shelah, S. (1984). Can you take Solovay’s inaccessible away? Israel Journal of Mathematics, 48, 1–47.
Solovay, R. M. (1965). The measure problem (abstract). Notices of the American Mathematical Society, 12, 217.
Solovay, R. M. (1969). The cardinality of \(\Sigma ^1_2\) sets of reals. In J.J. Bulloff, T.C. Holyoke, & S.W. Hahn (Eds.), Foundations of mathematics: Symposium papers commemorating the sixtieth birthday of Kurt Gödel (pp. 58–73). Berlin: Springer.
Solovay, R. M. (1970). A model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics, 92, 1–56.
Tait, W. W. (2005). Constructing cardinals from below. In W. W. Tait (Ed.), The provenance of pure reason: Essays in the philosophy of mathematics and its history (pp. 133–154). Oxford: Oxford University Press.
Wagon, S. (1994). The banach-tarski paradox. Cambridge: Cambridge University Press.
Wang, H. (1996). A logical journey: From Godel to philosophy. Boston: MIT Press.
Welch, P. D. (2012). Global reflection principles. Isaac Newton Institute Pre-print Series. Exploring the Frontiers of Incompleteness, EFI Workshop Papers, Harvard (INI12051-SAS).
Welch, P. D., & Horsten, L. (2016). Reflecting on absolute infinity. Journal of Philosophy, 113, 89–111.
Woodin, W. H. (1988). Supercompact cardinals, sets of reals, and weakly homogeneous trees. Proceedings of the National Academy of Sciences of the United States of America, 85(18), 6587–6591.
Woodin, W. H. (2001a). The continuum hypothesis, Part I. Notices of the American Mathematical Society, 48, 567–576.
Woodin, W. H. (2001b). The continuum hypothesis, Part II. Notices of the American Mathematical Society, 48, 681–690.
, chapter 22. In M. Magidor, M. Foreman, & A. Kanamori (Eds.), Handbook of set theory (vol. III). Berlin/New York: Springer.Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Welch, P.D. (2019). Proving Theorems from Reflection. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-15655-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-15654-1
Online ISBN: 978-3-030-15655-8
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)