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The non-compactness of square

  • James Cummings (a1), Matthew Foreman (a2) and Menachem Magidor (a3)

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This note proves two theorems. The first is that it is consistent to have for every n, but not have . This is done by carefully collapsing a supercompact cardinal and adding square sequences to each ωn. The crux of the proof is that in the resulting model every stationary subset of ℵω+1 ⋂ cof(ω) reflects to an ordinal of cofinality ω1, that is to say it has stationary intersection with such an ordinal.

This result contrasts with compactness properties of square shown in [3]. In that paper it is shown that if one has square at every ωn, then there is a square type sequence on the points of cofinality ωk, k > 1 in ℵω+1. In particular at points of cofinality greater than ω1 there is a strongly non-reflecting stationary set of points of countable cofinality.

The second result answers a question of Džamonja, by showing that there can be no squarelike sequence above a supercompact cardinal, where “squarelike” means that one replaces the requirement that the cofinal sets be closed and unbounded by the requirement that they be stationary at all points of uncountable cofinality.

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[1]Cummings, J., Collapsing successors of singulars, Proceedings of the American Mathematical Society, vol. 125 (1997), pp. 27032709.
[2]Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 3599.
[3]Cummings, J., Foreman, M., and Magidor, M., Canonical structure in the universe of set theory: Part I, to appear.
[4]Foreman, M., Games played on Boolean algebras, this Journal, vol. 48 (1983), pp. 714723.
[5]Foreman, M. and Magidor, M., Mutually stationary sequences and the non-saturation of the non-stationary ideal on P κ(λ), Acta Mathematica, vol. 186 (2001), no. 2.
[6]Laver, R., Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385388.
[7]Magidor, M., Foreman, M., and Shelah, S., Martin's maximum, saturated ideals and non-regular ultrafilters i, Annals of Mathematics, vol. 127 (1988), pp. 147.
[8]Namba, K., Independence proof of (ω, ω α)-distributive law in complete Boolean algebras, Commen-tarii Mathematici Universitatis Sancti Pauli, vol. 19 (1971), pp. 112.
[9]Namba, K., (ω 1, 2)-distributive law and perfect sets in generalized Baire space, Commentarii Mathematici Universitatis Sancti Pauli, vol. 20 (1971/1972), pp. 107126.
[10]Shelah, S., Proper Forcing, North-Holland, Amsterdam, 1980.

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