Abstract
We introduce a class of notions of forcing which we call Σ-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are Σ-Prikry. We show that given a Σ-Prikry poset and a name for a non-reflecting stationary set T, there exists a corresponding Σ-Prikry poset that projects to and kills the stationarity of T. Then, in a sequel to this paper, we develop an iteration scheme for Σ-Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If k is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which remains a strong limit cardinal, every finite collection of stationary subsets of K+ reflects simultaneously, and 2k = k++.
| Original language | English |
|---|---|
| Pages (from-to) | 1205-1238 |
| Number of pages | 34 |
| Journal | Canadian Journal of Mathematics |
| Volume | 73 |
| Issue number | 5 |
| DOIs | |
| State | Published - 26 Oct 2021 |
Bibliographical note
Publisher Copyright:© Canadian Mathematical Society 2020.
Funding
Poveda was partially supported by the Spanish Government under grant MTM2017-86777-P, by Generalitat de Catalunya (Catalan Government) under grant SGR 270-2017 and by MECD Grant FPU15/00026. Rinot was partially supported by the European Research Council (grant agreement ERC-2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18). Sinapova was partially supported by the National Science Foundation, Career-1454945.
| Funders | Funder number |
|---|---|
| Spanish Government | MTM2017-86777-P |
| National Science Foundation | |
| Horizon 2020 Framework Programme | 802756 |
| European Commission | |
| Generalitat de Catalunya | SGR 270-2017 |
| Ministerio de Educación, Cultura y Deporte | FPU15/00026 |
| Israel Science Foundation | 2066/18 |
Keywords
- Sigma-Prikry forcing
- Singular cardinals hypothesis
- Stationary reflection