Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T17:01:51.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorics on large cardinals

Published online by Cambridge University Press:  12 March 2014

Carlos H. Montenegro E.
Carlos H. Montenegro E.*
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125
*
Departamento de Matematicas, Universidad de los Andes, Bogotá, Colombia, E-mail: cmontene@andescol.bitnet
Get access Rights & Permissions [Opens in a new window]

Extract

Our framework is ZFC, and we view cardinals as initial ordinals. Baumgartner ([Bal] and [Ba2]) studied properties of large cardinals by considering these properties as properties of normal ideals and not as properties of cardinals alone. In this paper we study these combinatorial properties by defining operations which take as input one or more ideals and give as output an ideal associated with a large cardinal property. We consider four operations T, P, S and C on ideals of a regular cardinal κ, and study the structure of the collection of subsets they give, and the relationships between them.

The operation T is defined using combinatorial properties based on trees 〈X, <T〉 on subsets X ⊆ κ (where α <T β → α < β). Given an ideal I, consider the property *: “every tree on κ with every branching set in I has a branch of size κ” (where a branching set is a maximal set with the same set of <T-predecessors, and a chain is a maximal <T-linearly ordered set; for definitions see §2). Now consider the collection T(I) of all subsets of κ that do not satisfy * (see Definition 2.2 and the introduction to §5). The operation T provides us with the large cardinal property (whether κ ∈ T(I) or not) and it also provides us with the ideal associated with this large cardinal property (namely T(I)); in general, we obtain different notions depending on the ideal I.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 2 , June 1992 , pp. 617 - 643
Copyright
Copyright © Association for Symbolic Logic 1992

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

[Ba1]Baumgartner, J., Ineffability properties of cardinals, I, Infinite and finite sets (Keszthely, 1973, dedicated to P. Erdős on his sixtieth birthday), Vol. I, Colloquia Mathematica Societatis János Bolyai, vol. 10, North-Holland, Amsterdam, 1975, pp. 109–130.Google Scholar
[Ba2]Baumgartner, J., Ineffability properties of cardinals. II, Logic, foundations of mathematics and computability theory (proceedings of the fifth international congress of logic, methodology and philosophy of science, London, Ontario, 1975; Butts, R. and Hintikka, J., editors), Vol. I, Reidel, Dordrecht, 1977, pp. 87–106.Google Scholar
[BTW]Baumgartner, J., Taylor, A. and Wagon, S., Structural properties of ideals, Dissertationes Mathematical' (Rozpraway Matematyczne), vol. 197 (1982).Google Scholar
[Br]Barbanel, J. B., Supercompact cardinals, trees of normal ultrafilters, and the partition property, this Journal, vol. 51 (1986), pp. 701–714.Google Scholar
[Dr]Drake, F. R., Set theory: an introduction to large cardinals, North-Holland, Amsterdam, 1974.Google Scholar
[Fo]Fodor, G., Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Universitatis Szegedensis: Acta Scientiarum Mathematicarum, vol. 17 (1956), pp. 139–142.Google Scholar
[Ma]Matet, P., Some aspects of n-subtlety, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988), pp. 189–192.CrossRefGoogle Scholar
[Ne]Neumer, W., Verallgemeinerung eines Satzes von Alexandroff und Urysohn, Mathematische Zeitschrift, vol. 54 (1951), pp. 254–261.CrossRefGoogle Scholar