Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T13:14:28.622Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong axioms of infinity in NFU

Published online by Cambridge University Press:  12 March 2014

M. Randall Holmes
M. Randall Holmes*
Affiliation:
Department of Mathematics, Boise State University, Boise, Idaho 83725, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

This paper discusses a sequence of extensions of NFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].

The original theory NF of Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] that NF disproves Choice (and so proves Infinity). Even if one assumes the consistency of NF, one is hampered by the lack of powerful methods for proofs of consistency and independence such as are available for use with ZFC; very clever work has been done with permutation methods, starting with [18] and [5], and exemplified more recently by [14], but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition of NFU below for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent of NF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting from NF (see below for a statement of this axiom).

We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in [13]), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 87 - 116
Copyright
Copyright © Association for Symbolic Logic 2001

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

REFERENCES

[1]Boffa, M., ZFJ and the consistency problem for NF, Jahrbuch der Kurt Gödel Gesellschaft, 1988, pp. 102–106.Google Scholar
[2]Boffa, Maurice, Entre NF et NFU, Comptes Rendues de l'Academie des Sciences de Paris, series A, vol. 277 (1973), pp. 821–822.Google Scholar
[3]Forster, T. E., Set theory with a universal set, an exploration of an untyped universe, 2nd ed., Oxford logic guides, vol. 31, Oxford University Press, 1995.CrossRefGoogle Scholar
[4]Hailperin, T., A set of axioms for logic, this Journal, vol. 9 (1944), pp. 1–19.Google Scholar
[5]Henson, C. W., Permutation methods applied to NF, this Journal, vol. 38 (1973), pp. 69–76.Google Scholar
[6]Henson, C. W., Type-raising operations in NF, this Journal, vol. 38 (1973), pp. 59–68.Google Scholar
[7]Hinnion, Roland, Sur la théorie des ensembles de Quine, Ph. d. thesis, ULB Brussels, 1975.Google Scholar
[8]Hinnion, Roland, Stratified and positive comprehension seen as superclass rules over ordinary set theory, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 36 (1990), pp. 519–534.CrossRefGoogle Scholar
[9]Holmes, M. Randall, Systems of combinatory logic related to Quine's ‘New foundations’, Ph. d. thesis, SUNY Binghamton (now Binghamton University), 1990.Google Scholar
[10]Holmes, M. Randall, The set-theoretical program of Quine succeeded, but nobody noticed, Modern Logic, vol. 4 (1994), no. 1, pp. 1–47.Google Scholar
[11]Holmes, M. Randall, Elementary set theory with a universal set, Cahiers du Centre de logique, vol. 10, Academia, Louvain-la-Neuve, Belgium, 1998.Google Scholar
[12]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[13]Jensen, Ronald Bjorn, On the consistency of a slight (?) modification of Quine's ‘New foundations’, Synthese, vol. 19 (1969), pp. 250–263.CrossRefGoogle Scholar
[14]Körner, Friederike, Cofinal indiscernibles and some applications to New foundations, Mathematical Logic Quarterly, vol. 40 (1994), pp. 347–356.CrossRefGoogle Scholar
[15]Orey, S., New foundations and the axiom of counting, Duke Mathematical Journal, vol. 31 (1964), pp. 655–660.CrossRefGoogle Scholar
[16]Quine, W. V. O., New foundations for mathematical logic, American Mathematical Monthly, vol. 44 (1937), pp. 70–80.CrossRefGoogle Scholar
[17]Rosser, J. Barkley, Logic for mathematicians, 2nd ed., Chelsea, New York, 1973.Google Scholar
[18]Scott, Dana, Quine's individuals, Logic, methodology and philosophy of science (Nagel, E., editor), Stanford, 1962, pp. 111–115.Google Scholar
[19]Solovay, Robert, The consistency strength of NFUB, preprint, available through logic e-prints on the WWW.Google Scholar
[20]Specker, E. P., The axiom of choice in Quine's ‘New foundations for mathematical logic’, Proceedings of the National Academy of Sciences of the U. S. A., vol. 39 (1953), pp. 972–975.CrossRefGoogle Scholar
[21]Specker, E. P., Typical ambiguity, Logic, methodology, and philosophy of science (Nagel, E., editor), Stanford, 1962.Google Scholar
[22]Wang, Hao, Reflections on Kurt Gödel, MIT, 1987.Google Scholar