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$ \omega\sb 3\omega\sb 1\to (\omega\sb 3\omega\sb 1,3)\sp 2$ requires an inaccessible


Authors: Lee Stanley, Dan Velleman and Charles Morgan
Journal: Proc. Amer. Math. Soc. 111 (1991), 1105-1118
MSC: Primary 03E05; Secondary 03E35, 03E45, 03E55
DOI: https://doi.org/10.1090/S0002-9939-1991-1049849-X
MathSciNet review: 1049849
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Abstract: We show that if there is a simplified $ ({\omega _2},1)$-morass with linear limits and $ {2^{{\aleph _1}}} = {\aleph _2}$, then $ {\omega _3}{\omega _1} \nrightarrow {({\omega _3}{\omega _1},3)^2}$. Thus, assuming $ {2^{{\aleph _1}}} = {\aleph _2}$, this negative relation holds in $ V$ if both $ {\aleph _2}$ and $ {\aleph _3}$ are (successor cardinals)$ ^{L}$, since in this case, well-known arguments show there is a simplified $ ({\omega _2},1)$-morass with linear limits. The contrapositive is that, assuming $ {2^{{\aleph _1}}} = {\aleph _2}$, the positive relation holds only if either $ {\aleph _2}$ or $ {\aleph _3}$ is (inaccessible)$ ^{L}$.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1049849-X
Article copyright: © Copyright 1991 American Mathematical Society