Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T06:28:25.376Z Has data issue: false hasContentIssue false
  • English
  • Français

A very weak square principle

Published online by Cambridge University Press:  12 March 2014

Matthew Foreman  and
Menachem Magidor
Matthew Foreman
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92717, USA, E-mail: mforeman@math.uci.edu
Menachem Magidor
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel, E-mail: menachem@math.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L. This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is equivalent to a common combinatorial device, which we call a Jensen matrix. In the second section we show that our principle is consistent with a supercompact cardinal. In the third section of this paper we show that this principle is exactly equivalent to the statement that every torsion free Abelian group has a filtration into σ-balanced subgroups. In the fourth section of this paper we show that this principle fails if you assume the Chang's Conjecture:

In the fifth section of the paper we review the proofs that the various weak squares we consider are strictly decreasing in strength. Section 6 was added in an ad hoc manner after the rest of the paper was written, because the subject matter of Theorem 6.1 fit well with the rest of the paper. It deals with a principle dubbed “Not So Very Weak Square”, which appears close to Very Weak Square but turns out not to be equivalent.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 1 , March 1997 , pp. 175 - 196
Copyright
Copyright © Association for Symbolic Logic 1997

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[1] Baumgartner, J., unpublished work.Google Scholar
[2] Ben-David, S. and Magidor, M., The weak □ is really weaker than full □, this Journal, vol. 51 (1986), pp. 1029–1033.Google Scholar
[3] Burke, M. and Magidor, M., Shelah's pcf theory and its applications, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 207–254.CrossRefGoogle Scholar
[4] Foreman, M. and Magidor, M., Large cardinals and definable counterexamples to the continuum hypothesis, to appear in the Annals of Pure and Applied Logic.Google Scholar
[5] Fuchs, L. and Magidor, M., Butler groups of arbitrary cardinality, Israel Journal of Mathematics, vol. 84 (1993), pp. 239–263.CrossRefGoogle Scholar
[6] Hajnal, A., Juhasz, I., and Shelah, S., Splitting strongly almost disjoint families, Transactions of the American Mathematical Society, vol. 295 (1986), no. 1, pp. 369–387.CrossRefGoogle Scholar
[7] Hajnal, A., Juhasz, I., and Weiss, W., Partitioning the pairs and triples of topological spaces, Topology and its Applications, vol. 35 (1990), no. 2–3, pp. 177–184.CrossRefGoogle Scholar
[8] Jensen, R., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), no. 3, pp. 229–308.CrossRefGoogle Scholar
[9] Levinski, J. P., Magidor, M., and Shelah, S., On Chang's conjecture for ℵω, Israel Journal of Mathematics, vol. 69 (1990), pp. 161–172.CrossRefGoogle Scholar
[10] Magidor, M., Reflecting stationary sets, this Journal, vol. 47 (1982), pp. 755–771.Google Scholar
[11] Magidor, M. and Shelah, S., When does almost free imply free?, to appear in the Journal of the American Mathematical Society.Google Scholar
[12] Shelah, S., On successors of singular cardinals, Logic colloquium '78 (Boffa, M.et al., editors), North-Holland, Amsterdam, 1979, pp. 357–380.Google Scholar