Chains of P-points

Raghavan, Dilip; Verner, Jonathan L.

doi:10.4153/s0008439519000043

Cited by 4 publications

(2 citation statements)

References 18 publications

(19 reference statements)

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“…A recent paper by D. Raghavan and J. L. Verner[9] showed that under ♦, the cardinal 1 is the answer, but we still do not know the answer in terms of cardinal invariants which, as we suppose, play an important role in this domain.…”

mentioning

confidence: 72%

THE RUDIN–KEISLER ORDERING OF P-POINTS UNDER 𝔟 = 𝔠

Starosolski¹

2021

J. symb. log.

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M. E. Rudin (1971) proved, under CH, that for each P-point p there exists a P-point q strictly RK-greater than p. This result was proved under ${\mathfrak {p}= \mathfrak {c}}$ by A. Blass (1973), who also showed that each RK-increasing $ \omega $ -sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-ordering. In this paper, the results cited above are proved under the (weaker) assumption that $\mathfrak { b}=\mathfrak {c}$ . A. Blass asked in 1973 which ordinals can be embedded in the set of P-points, and pointed out that such an ordinal cannot be greater than $ \mathfrak {c}^{+}$ . In this paper it is proved, under $\mathfrak {b}=\mathfrak {c}$ , that for each ordinal $\alpha < \mathfrak {c}^{+}$ , there is an order embedding of $ \alpha $ into P-points. It is also proved, under $\mathfrak {b}=\mathfrak {c}$ , that there is an embedding of the long line into P-points.

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