More consistency results in partition calculus |
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Authors: | Saharon Shelah Lee Stanley |
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Institution: | (1) Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel;(2) Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA;(3) Department of Mathematics, Lehigh University, 18015 Bethlehem, PA, USA |
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Abstract: | This paper has two principal aims. The first is to supply a proof of Theorem 6 of ShSt1]:Theorem:If ZFC+ “there are c
+
measurable cardinals” is consistent, then so is ZFC+ “ ℵ
c
+is not a strong limit cardinal and
”.
This is done in sections 1 and 2. See the introduction for a discussion of the evolution of the proof and of some interesting
questions which remain open, related to obstacles encountered in obtaining maximum freedom in arranging for any desired cardinal
exponentiation in Theorems 4 and 6 of ShSt1]. The method is quite generally applicable in partition calculus and variants
of it have in fact been applied in recent work of the authors, see ShSt2]. first, a preservation result is proved for the
game-theoretic properties of the filters considered in ShSt1]. Then, it is shown that the existence of a system of such filters
yields a canonization-style result. Finally, it is shown that the canonization property gives the positive partition relation.
The second aim makes the title of this paper slightly inaccurate (but we suspect this will be pardoned): we supply a (straightforward)
proof of a result which shows that Theorem 2 of ShSt1] in some sense is best possible. This is done in section 3.
Research of both authors partially supported by NSF grants.
Partially supported by the BSF. Paper number 293. |
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