The Consistency Strength of $$\aleph_{\omega}$$ and $$\aleph_{{\omega}_1}$$ Being Rowbottom Cardinals Without the Axiom of Choice

We show that for all natural numbers n, the theory “ZF + DC $$_{\aleph_n}$$ + $$\aleph_{\omega}$$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $$\aleph_{\omega_1}$$ is an ω2-Rowbottom cardinal carrying an ω2-Rowbottom filter and ω1 is regular” has the same consistency strength as the theory “ZFC + There exist ω1 measurable cardinals”. We also discuss some generalizations of these results.

[1]  A. Dodd.  The Core Model: INDEX OF DEFINITIONS , 1982 .

[2]  R. Jensen,et al.  The Core Model , 1982, An Introduction to Geographical and Urban Economics.

[3]  Daniel W. Cunningham.  A covering lemma for L(ℝ) , 2002, Arch. Math. Log..

[4]  Ralf Schindler.  Successive Weakly Compact or Singular Cardinals , 1999, J. Symb. Log..

[5]  Peter Koepke.  Some applications of short core models , 1988, Ann. Pure Appl. Log..

[6]  A. J. Dodd,et al.  The covering lemma for L[U] , 1982, Ann. Math. Log..

[7]  Jack Howard Silver,et al.  Some applications of model theory in set theory , 1966 .

[8]  Arthur W. Apter.  On a Problem of Silver , 1983 .

[9]  Arthur W. Apter.  On a problem of Woodin , 2000, Arch. Math. Log..

[10]  Arthur W. Apter.  AD and patterns of singular cardinals below Θ , 1996, Journal of Symbolic Logic.

[11]  Peter Koepke,et al.  The consistency strength of the free-subset property for ωω , 1984, Journal of Symbolic Logic.

[12]  A. Levy,et al.  Measurable cardinals and the continuum hypothesis , 1967 .