The *-Prikry condition

Vincenzo Dimonte


In this paper we isolate a property for forcing notions, the *-Prikry condition, that is similar to the Prikry condition but that is topological: A forcing P satisfies it iff for every p ∈Pand for every open dense D ⊆P, there are n ∈ωand q ≤∗p such that for any rq with l(r) = l(q) + n, rD, for some length notion l. This is implicit in many proofs in literature. We prove this for the tree Prikry forcing and the long extender Prikry forcing.


Received 16 October 2017

Revised 2 June 2018

AMS subject classifications: 03E55, 03E05, 03E35(03E45)

Słowa kluczowe: Prikry forcing

[1] S. Cramer, Implications of very large cardinals, Contemporary Mathematics 690 (2017), 225–257.

[2] V. Dimonte, Totally non-proper ordinals beyond L(Vλ+1), Archive for Mathematical Logic 50 (2011), 565–584.

[3] V. Dimonte and S.-D. Friedman, Rank-into-rank hypotheses and the failure of GCH, Archive for Mathematical Logic 53 (2014), 351–366.

[4] V. Dimonte and L. Wu, A general tool for consistency results related to I1, European Journal of Mathematics 2 (2016), 474–492.

[5] M. Gitik, Prikry-type forcings, vol. 2 of Handbook of set theory, Springer, Dordrecht, 2010, pp. 1351–1447.

[6] M. Gitik, Short extender forcings I, Journal of Mathematical Logic 12:2 (2012).

[7] M. Gitik and A. Sharon, On SCH and the approachability property, Proceedings of the American Mathematical Society 136 (2008), 311–320.

[8] C. Merimovich, Prikry on extenders, revisited, Israel Journal of Mathematics 160 (2007), 253–280.

[9] C. Merimovich, Supercompact extender based Prikry forcing, Archive for Mathematical Logic 50 (2011), 591–602.

[10] I. Neeman, Aronszajn trees and failure of the Singular Cardinal Hypothesis, Journal of Mathematical Logic 9 (2009), 139–157.

[11] F. Rowbottom, Some strong axioms of infinity incompatible with the axiom of constructibility, Annals of Mathematical Logic 3 (1971), 1–44.

[12] D. Scott, Measurable cardinals and constructible sets, Bulletin de l’Acad´emie Polonaise des Sciences, S´erie des Sciences Math´ematiques, Astronomiques et Physiques 9 (1961), 521–524.

[13] S. Shelah, On nice equivalence relations on λ2, Archive for Mathematical Logic 43 (2004), 31–64.

[14] X. Shi, Axiom I0 and higher degree theory, The Journal of Symbolic Logic 80 (2015), 970–1021.

[15] X. Shi and N. Trang, I0 and combinatorics at λ+, Archive for Mathematical Logic 56 (2017), 131–154.

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