Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T19:31:30.839Z Has data issue: false hasContentIssue false
  • English
  • Français

Rowbottom cardinals and Jonsson cardinals are almost the same1

Published online by Cambridge University Press:  12 March 2014

E. M. Kleinberg
E. M. Kleinberg*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Get access Rights & Permissions [Opens in a new window]

Extract

Each of the various “large cardinal” axioms currently studied in set theory owes its inspiration to concrete phenomena in various fields. For example, the statement of the well-known compactness theorem for first-order logic can be generalized in various ways to infinitary languages to yield definitions of compact cardinals, and the reflection principles provable in ZF, when modified in the appropriate way, yields indescribable cardinals.

In this paper we concern ourselves with two kinds of large cardinals which are probably the two best known of those whose origins lie in model theory. They are the Rowbottom cardinals and the Jonsson cardinals.

Let us be more specific. A cardinal κ is said to be a Jonsson cardinal if every structure of cardinality κ has a proper elementary substructure of cardinality κ. (It is routine to see that only uncountable cardinals can be Jonsson. Erdös and Hajnal have shown [2] that for n < ω no ℵn is Jonsson. (In fact, they showed that if κ is not Jonsson then neither is the successor cardinal of κ and that, assuming GCH, no successor cardinal can be Jonsson.) Keisler and Rowbottom first showed that the existence of a Jonsson cardinal contradicts V = L.) The definition of a Rowbottom cardinal is only slightly more intricate. We assume for the moment that our similarity type has a designated one-place relation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 3 , September 1973 , pp. 423 - 427
Copyright
Copyright © Association for Symbolic Logic 1973

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.)

Footnotes

1

This research was partially supported by NSF grant GP-29079.

References

REFERENCES

[1]Devlin, K., Some weak versions of large cardinal axioms, Doctoral Dissertation, University of Bristol, 1971.Google Scholar
[2]Erdös, P. and Hajnal, A., On a problem of B. Jonsson, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 14 (1966), pp. 19–23.Google Scholar
[3]Kleinberg, E. M., Rowbottom cardinals and Jonsson cardinals are almost the same, Notices of the American Mathematical Society, vol. 18 (1971), p. 827. Abstract 71T-E85.Google Scholar
[4]Kleinberg, E. M., The equiconsistency of two large cardinal axioms (to appear).Google Scholar
[5]Kleinberg, E. M., The abstract compactness idea, M.I.T. Logic Seminar, 1971–1972 (to appear).Google Scholar
[6]Rowbottom, F., Some strong axioms of infinity incompatible with the axiom of constructibility, Annals of Mathematical Logic, vol. 3 (1971), pp. 1–44.CrossRefGoogle Scholar
[7]Silver, J., Some applications of model theory in set theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 45–110.CrossRefGoogle Scholar