A consistent consequence of AD

Bull, E. L.; Kleinberg, E. M.

doi:10.1090/S0002-9947-1979-0517692-2

A consistent consequence of AD
HTML articles powered by AMS MathViewer

by E. L. Bull and E. M. Kleinberg

Trans. Amer. Math. Soc. 247 (1979), 211-226

DOI: https://doi.org/10.1090/S0002-9947-1979-0517692-2

PDF | Request permission

Abstract:

We are concerned with the existence of a normal measure concentrating on the $\omega$-closed unbounded sets. We strengthen the known result that the axiom of choice implies no such measure exists. It is shown that the existence of these measures is consistent (relative to a large cardinal). In particular, ${\aleph _2}$ may admit exactly two normal measures: one which contains the $\omega$-closed unbounded sets and the other, the ${\aleph _1}$-closed unbounded sets. This property of ${\aleph _2}$ is a well-known consequence of AD.

References

Menachem Magidor, How large is the first strongly compact cardinal? or A study on identity crises, Ann. Math. Logic 10 (1976), no. 1, 33–57. MR 429566, DOI 10.1016/0003-4843(76)90024-3

has exactly two normal measures

William J. Mitchell, Sets constructible from sequences of ultrafilters, J. Symbolic Logic 39 (1974), 57–66. MR 344123, DOI 10.2307/2272343
K. L. Prikry, Changing measurable into accessible cardinals, Dissertationes Math. (Rozprawy Mat.) 68 (1970), 55. MR 262075
J. R. Shoenfield, Unramified forcing, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 357–381. MR 0280359
A. Lévy and R. M. Solovay, Measurable cardinals and the continuum hypothesis, Israel J. Math. 5 (1967), 234–248. MR 224458, DOI 10.1007/BF02771612