Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-18T14:59:05.114Z Has data issue: false hasContentIssue false

LEVEL THEORY, PART 1: AXIOMATIZING THE BARE IDEA OF A CUMULATIVE HIERARCHY OF SETS

Published online by Cambridge University Press:  06 May 2021

TIM BUTTON*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY COLLEGE LONDON GOWER STREET, LONDON, WC1E 6BT, UK E-mail: tim.button@ucl.ac.uk URL: http://www.nottub.com

Abstract

The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: ‘Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S’. Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, Montague, Derrick, and Potter.

Type
Articles
Copyright
The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boolos, G., The iterative conception of set . The Journal of Philosophy , vol. 68 (1971), no. 8, pp. 215231.10.2307/2025204CrossRefGoogle Scholar
Boolos, G., Iteration again . Philosophical Topics , vol. 17 (1989), no. 2, pp. 521.10.5840/philtopics19891721CrossRefGoogle Scholar
Burgess, J.P., E pluribus unum: Plural logic and set theory . Philosophia Mathematica , vol. 12 (2004), pp. 193221.10.1093/philmat/12.3.193CrossRefGoogle Scholar
Button, T. and Walsh, S., Philosophy and Model Theory , Oxford University Press, Oxford, 2018.CrossRefGoogle Scholar
Doets, K., Relatives of the Russell paradox . Mathematical Logic Quarterly , vol. 45 (1999), no. 1, pp. 7383.10.1002/malq.19990450107CrossRefGoogle Scholar
Drake, F.R., Set Theory: An Introduction to Large Cardinals , North-Holland, London, 1974.Google Scholar
Incurvati, L., How to be a minimalist about sets . Philosophical Studies , vol. 159 (2012), no. 1, pp. 6987.10.1007/s11098-010-9690-1CrossRefGoogle Scholar
Incurvati, L., Conceptions of Set and the Foundations of Mathematics, Cambridge University Press, Cambridge, 2020.CrossRefGoogle Scholar
Lévy, A. and Vaught, R., Principles of partial reflection in the set theories of Zermelo and Ackermann . Pacific Journal of Mathematics , vol. 11 (1961), no. 3, pp. 10451062.10.2140/pjm.1961.11.1045CrossRefGoogle Scholar
Mathias, A.R.D., Slim models of Zermelo set theory . The Journal of Symbolic Logic, vol. 66 (2001), no. 2, pp. 487496.CrossRefGoogle Scholar
McGee, V., Two problems with Tarski’s theory of consequence. Proceedings of the Aristotelian Society, vol. 92 (1992), pp. 273292.10.1093/aristotelian/92.1.273CrossRefGoogle Scholar
McGee, V., How we learn mathematical language . Philosophical Review , vol. 106 (1997), no. 1, pp. 3568.10.2307/2998341CrossRefGoogle Scholar
Menzel, C., On the iterative explanation of the paradoxes . Philosophical Studies , vol. 49 (1986), no. 1, pp. 3761.10.1007/BF00372882CrossRefGoogle Scholar
Menzel, C., Wide sets, ZFCU, and the iterative conception . The Journal of Philosophy , vol. 111 (2014), no. 2, pp. 5783.CrossRefGoogle Scholar
Montague, R., On the paradox of grounded classes . The Journal of Symbolic Logic , vol. 20 (1955), no. 2, p. 140.10.2307/2266899CrossRefGoogle Scholar
Montague, R. , Set theory and higher-order logic , Formal Systems and Recursive Functions: Proceedings of the Eighth Logic Colloquium, July 1963 (J. Crossley and M. Dummett, editors), North-Holland, Amsterdam, 1965, pp. 131148.10.1016/S0049-237X(08)71686-0CrossRefGoogle Scholar
Montague, R., Scott, D., and Tarski, A., An axiomatic approach to set theory , Archive copy from the Bancroft Library (BANC MSS 84/69 c, carton 4, folder 29–30), unpublished manuscript, n.d. Google Scholar
Parsons, C., The uniqueness of the natural numbers . Iyyun , vol. 39 (1990), no. 1, pp. 1344.Google Scholar
Parsons, C., Mathematical Thought and Its Objects, Harvard University Press, Cambridge, MA, 2008.Google Scholar
Potter, M., Sets: An Introduction , Oxford University Press, Oxford, 1990.Google Scholar
Potter, M., Iterative set theory. Philosophical Quarterly, vol. 43 (1993), no. 171, pp. 178193.10.2307/2220368CrossRefGoogle Scholar
Potter, M., Set Theory and Its Philosophy, Oxford University Press, Oxford, 2004.10.1093/acprof:oso/9780199269730.001.0001CrossRefGoogle Scholar
Rumfitt, I., The Boundary Stones of Thought: An Essay in the Philosophy of Logic, Oxford University Press, Oxford, 2015.10.1093/acprof:oso/9780198733638.001.0001CrossRefGoogle Scholar
Scott, D., The notion of rank in set-theory, Summaries of Talks Presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, Institute for Defence Analysis, Princeton, NJ, 1960, pp. 267269.Google Scholar
Scott, D., Axiomatizing set theory, Axiomatic Set Theory II (T. Jech, editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1974, pp. 207214.10.1090/pspum/013.2/0392570CrossRefGoogle Scholar
Shoenfield, J.R., Mathematical Logic, Addison-Wesley, London, UK, 1967.Google Scholar
Shoenfield, J.R., Axioms of set theory, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, London, 1977, pp. 321344.10.1016/S0049-237X(08)71106-6CrossRefGoogle Scholar
Uzquiano, G., Models of second-order Zermelo set theory, this Journal, vol. 5 (1999), no. 3, pp. 289302.Google Scholar
Uzquiano, G., A neglected resolution of Russell’s paradox of propositions . The Review of Symbolic Logic , vol. 8 (2015), pp. 328344.10.1017/S1755020315000106CrossRefGoogle Scholar
Väänänen, J. and Wang, T., Internal categoricity in arithmetic and set theory . Notre Dame Journal of Formal Logic , vol. 56 (2015), no. 1, pp. 121134.10.1215/00294527-2835038CrossRefGoogle Scholar
Zermelo, E., Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicæ, vol. 16 (1930), pp. 2947.10.4064/fm-16-1-29-47CrossRefGoogle Scholar