We show that the notion of cardinality of a set is independent from that of wellordering, and that reasonable total notions of cardinality exist in every model of ZF where the axiom of choice fails. Such notions are either definable in a simple and natural way, or non-definable, produced by forcing. Analogous cardinality notions exist in nonstandard models of arithmetic admitting nontrivial automorphisms. Certain motivating phenomena from quantum mechanics are also discussed in the Appendix.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
CHUAQUI, R., ‘Forcing for the impredicative theory of classes’, J. Symb.Logic 37, 1 (1972), 1–18.
P.COHEN, ‘Automorphisms of set theory’, Proc. Symp. Pure Mathematics,volume 25, AMS Providence 1974, pp. 325–330. Cardinality without Enumeration 141
DALLA CHIARA, M.L., R. GIUNTINI, and D. KRAUSE, ‘Quasiset theory for microobjects’, in: Elena Castelani, (ed.), Interpreting Bodies, Classical and Quantum Objects in Modern Physics, Princeton U.P. 1998, pp. 142-152.
DI FRANCIA, GIULIANO TORALDO, ‘A world of individual objects?’, in Elena Castelani, (ed.), Interpreting Bodies, Classical and Quantum Objects in Modern Physics, Princeton U.P. 1998, pp. 21–29.
HELLMAN, G., ‘Constructive mathematics and quantum mechanics: Unbounded operators and the spectral theorem’, J. Phil. Logic 22 (1993), 221–248.
JECH, T., The axiom of choice, North Holland 1973.
JECH, T., Set Theory, Springer Verlag 2002.
KAY, R., Models of Peano Arithmetic, Oxford Logic Guides, 1991.
KIRBY, L., K. MCALOON, and R. MURAWSKI, ‘Indicators, recursive saturation and expandability’, Fund. Math. CXIV (1981), 127–139.
KRAUSE, D., ‘On a quasi-set theory’, Notre Dame Journal of Formal Logic 33 (1992), 402–411.
KUNEN, K., Set theory, an introduction to independence proofs, NorthHolland, 1980.
LINDENBAUM, A., and A. TARSKI, Communications sur les recherces de laThéorie des Ensebles, in Steven R. Givant and Ralph N. McKenzie, (eds.), Alfred Tarski Collected Papers, Volume I, Birkhauser 1986, pp. 171-204.
MYHILL, J., ‘Constructive set theory’, J. Symb. Logic 40 (1975), 347–382.
MYHILL, J., ‘Intensional set theory’, in S. Shapiro, (ed.), Intensional mathematics, North Holland 1985, pp. 47–61.
REICHENBACH, HANS, ‘The genidentity of quantum particles’, in Elena Castelani, (ed.), Interpreting Bodies, Classical and Quantum Objects in Modern Physics, Princeton U.P. 1998, pp. 61–72.
RUBIN, H., and J. RUBIN, Equivalents of the Axiom of Choice, North Holland, 1963.
SCHMERL, J., ‘Peano models with many generic classes’, Pacific J. Math. 46, 2 (1973), 523–536.
TELLER, PAUL, ‘Quantum Mechanics and Haecceities’, in Elena Castelani, (ed.), Interpreting Bodies, Classical and Quantum Objects in Modern Physics, Princeton U.P. 1998, pp. 114–141.
ZARACH, A., ‘Forcing with proper classes’, Fund. Math. LXXXI (1973), 1–27.
About this article
Cite this article
Tzouvaras, A. Cardinality without Enumeration. Stud Logica 80, 121–141 (2005). https://doi.org/10.1007/s11225-005-6780-8
- Cardinality notion
- generic class
- separation class
- nonstandard model of arithmetic
- quantum mechanics