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A very weak square principle

  • Matthew Foreman (a1) and Menachem Magidor (a2)

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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.

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[1] Baumgartner, J., unpublished work.
[2] Ben-David, S. and Magidor, M., The weak □ is really weaker than full □, this Journal, vol. 51 (1986), pp. 10291033.
[3] Burke, M. and Magidor, M., Shelah's pcf theory and its applications, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 207254.
[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.
[5] Fuchs, L. and Magidor, M., Butler groups of arbitrary cardinality, Israel Journal of Mathematics, vol. 84 (1993), pp. 239263.
[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. 369387.
[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. 177184.
[8] Jensen, R., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), no. 3, pp. 229308.
[9] Levinski, J. P., Magidor, M., and Shelah, S., On Chang's conjecture for ℵω, Israel Journal of Mathematics, vol. 69 (1990), pp. 161172.
[10] Magidor, M., Reflecting stationary sets, this Journal, vol. 47 (1982), pp. 755771.
[11] Magidor, M. and Shelah, S., When does almost free imply free?, to appear in the Journal of the American Mathematical Society.
[12] Shelah, S., On successors of singular cardinals, Logic colloquium '78 (Boffa, M.et al., editors), North-Holland, Amsterdam, 1979, pp. 357380.

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