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 satis es it i for every p 2 P and for every open dense D  P, there are n 2 ! and q  p such that for any r  q with l(r) = l(q) + n, r 2 D, 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.

Słowa kluczowe: Prikry forcing

S. Cramer, Implications of very large cardinals, Contemporary Mathematics 690 (2017), 225-257.

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

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

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

M. Gitik, Prikry-type forcings, Handbook of set theory, Vol. 2, Springer, Dordrecht 2010, pp. 1351-1447.

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

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

C. Merimovich, Prikry on extenders, revisited, Israel Journal of Mathematics 160 (2007), 253-280.

C. Merimovich, Supercompact extender based Prikry forcing, Archive for Mathematical Logic 50 (2011), 591-602.

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

F. Rowbottom, Some strong axioms of in nity incompatible with the axiom of constructibility, Annals of Mathematical Logic 3 (1971), 1-44.

D. Scott, Measurable cardinals and constructible sets, Bulletin de l'Academie Polonaise des Sciences, Serie des Sciences Mathematiques, Astronomiques et Physiques 9 (1961), 521-524.

S. Shelah, On nice equivalence relations on 2, Archive for Mathematical Logic 43 (2004), 31-64.

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

X. Shi, N. Trang, I0 and combinatorics at +, Archive for Mathematical Logic 56 (2017), 131-154.

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