Weakly compact cardinals and nonspecial Aronszajn trees

Authors:
Saharon Shelah and Lee Stanley

Journal:
Proc. Amer. Math. Soc. **104** (1988), 887-897

MSC:
Primary 03E05; Secondary 03E45, 03E55, 04A20

DOI:
https://doi.org/10.1090/S0002-9939-1988-0964870-5

MathSciNet review:
964870

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Abstract: Lemma 1. *If* *is a cardinal with* cf , *then* *implies that there is a* *-Aronszajn tree with an* *-ascent path, i.e. a sequence* *with each* *a one-to-one sequence from* , *such that for all* *precedes* *in the tree order for sufficiently large* .

Lemma 2. *If* *is a cardinal with* , *then* *implies that there is a* *-Aronszajn tree with an* *-ascent path* (*replace* *by* , *above*).

Lemma 3. *If* *is an uncountable cardinal*, *is regular*, is a *-Aronszajn tree, and* *is a one-to-one sequence from* *with the property of ascent paths, where* *is a monotone increasing function of* , *then* *is nonspecial*.

Theorem 4. *If* *is uncountable, then* *implies that there is a nonspecial* *-Aronszajn tree*.

Theorem 5. *If* *is an uncountable cardinal*, , *and* *is not* , *then there is a nonspecial* *-Aronszajn tree*.

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0964870-5

Article copyright:
© Copyright 1988
American Mathematical Society