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Some results on consecutive large cardinals. II: Applications of Radin forcing. (English) Zbl 0603.03016
[For Part I see Ann. Pure Appl. Logic 25, 1-17 (1983; Zbl 0548.03030).]
Let $$\kappa$$ be a 3 huge cardinal in a countable model V of ZFC, and let A and B be subsets of the successor ordinals $$<\kappa$$ so that $$A\cup B=\{\alpha <\kappa:\alpha$$ is a successor ordinal$$\}$$. Using techniques of Gitik, we construct a choiceless model $$N_ A$$ of ZF of height $$\kappa$$ so that $$N_ A\vDash ''ZF + \neg AC_{\omega} + For\alpha\in A$$, $$\aleph_{\alpha}$$ is a Ramsey cardinal $$+ For\beta\in B$$, $$\aleph_{\beta}$$ is a singular Rowbottom cardinal which carries a Rowbottom filter $$+ For\gamma$$ a limit ordinal, $$\aleph_{\gamma}$$ is a Jonsson cardinal which carries a Jonsson filter”.

##### MSC:
 3e+55 Large cardinals
Full Text:
##### References:
 [1] A. Apter,Some results on consecutive large cardinals, Ann. Pure Appl. Logic25 (1983), 1–17. · Zbl 0548.03030 [2] E. Bull,Successive large cardinals, Ann. Math. Logic15 (1978), 161–191. · Zbl 0402.03048 [3] M. Foreman and H. Woodin,The GCH can fail everywhere, to appear. · Zbl 0718.03040 [4] M. Gitik,Regular cardinals in models of ZF, Trans. Am. Math. Soc., to appear. · Zbl 0589.03033 [5] A. Kanamori and M. Magidor,The evolution of large cardinal axioms in set theory, Lecture Notes in Mathematics685, Springer-Verlag, Berlin, 1979. · Zbl 0381.03038 [6] E. Kleinberg,Infinitary Combinatorics and the Axiom of Determinateness, Lecture Notes in Mathematics612, Springer-Verlag, Berlin, 1977. · Zbl 0362.02067 [7] R. Laver,Making the supercompactness of {$$\kappa$$} indestructible under {$$\kappa$$} directed closed forcing, Israel J. Math.29 (1978), 383–388. · Zbl 0381.03039 [8] A. Lévy and R. Solovay,Measurable cardinals and the Continuum Hypothesis, Israel J. Math.5 (1967), 234–248. · Zbl 0289.02044 [9] T. Menas, A combinatorial property ofP K({$$\lambda$$}), J. Symbolic Logic41 (1975), 225–233. [10] T. Menas,On strong compactness and supercompactness, Ann. Math. Logic7 (1975), 327–359. · Zbl 0299.02084 [11] W. Mitchell,How weak is a closed unbounded filter?, Logic Colloq. ’80 (Van Dalen, Lascar and Smiley, eds.), North-Holland, 1982. · Zbl 0496.03031 [12] K. Prikry,Changing measurable into accessible cardinals, Dissertationes Math. (Rozprany Mathematyczne)68 (1970), 5–52. · Zbl 0212.32404 [13] L. Radin,Adding closed cofinal sequences to large cardinals, Ann. Math. Logic23 (1982), 263–283. · Zbl 0502.03028 [14] F. Rowbottom,Some strong axioms of infinity incompatible with the axiom of constructibility, Ann. Math. Logic3 (1971), 1–44. · Zbl 0274.02034 [15] H. Woodin, Handwritten notes on the closed unbounded filter. [16] H. Woodin, Handwritten notes on Radin forcing and the Prikry property.
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