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Universal ${ \aleph }_{2}$-Aronszajn trees

By Mirna Dzamonja

Appears in collection : XVI International Luminy Workshop in Set Theory / XVI Atelier international de théorie des ensembles

We report on a joint work in progress with Rahman Mohammadpour in which we study the problem of the possible existence of a universal tree under weak embeddings in the classes of $\aleph_{2}$-Aronszajn and wide $\aleph_{2}$-Aronszajn trees. This problem is more complex than previously thought, in particular it seems not to be resolved under ShFA $+$ CH using the technology of weakly Lipshitz trees. We show that under CH, for a given $\aleph_{2}$-Aronszajn tree $\mathrm{T}$ without a weak ascent path, there is an $\aleph_{2^{-\mathrm{C}\mathrm{C}}}$ countably closed forcing forcing which specialises $\mathrm{T}$ and adds an $\aleph_{2}$-Aronszajn tree which does not embed into T. One cannot however apply the ShFA to this forcing.

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  • DOI 10.24350/CIRM.V.19809103
  • Cite this video Dzamonja Mirna (9/16/21). Universal ${ \aleph }_{2}$-Aronszajn trees. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19809103
  • URL https://dx.doi.org/10.24350/CIRM.V.19809103

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