Forcing the $\Pi^1_n$-Uniformization Property

Stefan Hoffelner

Forcing the $\Pi^1_n$-Uniformization Property

Logic 2022-10-18 v4

Authors: Stefan Hoffelner

Abstract

We generically construct a model in which the ${\Pi^1_3}$ -uniformization property is true, thus lowering the best known consistency strength from the existence of $M_1^{\#}$ to just $\mathsf{ZFC}$ . The forcing construction can be adapted to work over canonical inner models with Woodin cardinals, which yields, for the first time, universes where the $\Pi^1_{2n}$ -uniformization property holds for $n >1$ , thus producing models which contradict the natural $\mathsf{PD}$ -induced pattern. It can also be used to obtain models for the $\Pi^1_1$ -uniformization property in the generalized Baire space.

Keywords

set theory forcing

Cite

@article{arxiv.2103.11748,
  title  = {Forcing the $\Pi^1_n$-Uniformization Property},
  author = {Stefan Hoffelner},
  journal= {arXiv preprint arXiv:2103.11748},
  year   = {2022}
}

Comments

63 pages, slightly altered coding method which makes several definitions more transparent while leaving the general flow of ideas and proofs untouched. arXiv admin note: text overlap with arXiv:2009.02209, arXiv:1912.11811

Forcing the $\Pi^1_n$-Uniformization Property

Abstract

Keywords

Cite

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