A Side Condition Principle With Applications
Abstract
We introduce a combinatorial principle textsfSC(κ), for any regular cardinal κ≥ ω2, which asserts the existence of a family 𝒴 ⊆ [H(κ)]ω of elementary substructures of H(κ) which is closed under finite intersections and satisfies that for stationarily many N ∈ 𝒴, if M ∈ 𝒴 and M ∩ ω1 < N ∩ ω1, then M ∩ N ∩ κ∈ N. We explore the idea of defining forcings with side conditions consisting of finite subsets of such a family 𝒴. If textsfSC(ω2) holds, then there exists a thin stationary subset of [ω2]ω, and consequently textsfSC(ω2) is independent of ZFC. Assuming that μ is a Mahlo cardinal, the Lévy collapse Col(ω,<μ) forces that textsfSC(κ) holds for all regular cardinals κ≥ ω2. Local instances of textsfSC can be forced using matrices of models or derived as consequences of square principles. We apply the principle textsfSC to define two kinds of forcings: a streamlined version of a strongly proper collapse, and a forcing which is strongly proper on a stationary set and adds arbitrarily many Baumgartner clubs. Finally, we prove two consistency results using forcings derived from textsfSC related to Aronszajn trees, the guessing model principle, strong failures of club guessing, and the higher dominating number.
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BibTeX
@misc{jakob2026side,
title = {{A Side Condition Principle With Applications}},
author = {Hannes Jakob and John Krueger},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04362},
eprint = {2611.04362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04362},
}
RIS
TY - UNPB
TI - A Side Condition Principle With Applications
AU - Hannes Jakob
AU - John Krueger
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04362
UR - https://arxiv.org/abs/2611.04362
AB - We introduce a combinatorial principle textsfSC(κ), for any regular cardinal κ≥ ω_2, which asserts the existence of a family 𝒴 ⊆ [H(κ)]^ω of elementary substructures of H(κ) which is closed under finite intersections and satisfies that for stationarily many N ∈ 𝒴, if M ∈ 𝒴 and M ∩ ω_1 < N ∩ ω_1, then M ∩ N ∩ κ∈ N. We explore the idea of defining forcings with side conditions consisting of finite subsets of such a family 𝒴. If textsfSC(ω_2) holds, then there exists a thin stationary subset of [ω_2]^ω, and consequently textsfSC(ω_2) is independent of ZFC. Assuming that μ is a Mahlo cardinal, the Lévy collapse Col(ω,<μ) forces that textsfSC(κ) holds for all regular cardinals κ≥ ω_2. Local instances of textsfSC can be forced using matrices of models or derived as consequences of square principles. We apply the principle textsfSC to define two kinds of forcings: a streamlined version of a strongly proper collapse, and a forcing which is strongly proper on a stationary set and adds arbitrarily many Baumgartner clubs. Finally, we prove two consistency results using forcings derived from textsfSC related to Aronszajn trees, the guessing model principle, strong failures of club guessing, and the higher dominating number.
ER -