Forcing with Invariant Measures
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11787_2025_Article_394.pdf
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Author(s)
Ackerman, Nathanael
Freer, Cameron
Golshani, Mohammad
Mirabi, Mostafa
Patel, Rehana
Date Issued
November 24, 2025
Journal
Logica Universalis
Publisher
Springer International Publishing
Citation
Ackerman, N., Freer, C., Golshani, M. et al. Forcing with Invariant Measures. Log. Univers. (2025).
Version
Final published version
Abstract
This paper introduces a model-theoretic generalization of the notion of forcing with random reals, in which forcing gives rise to random generic structures. Specifically, we consider forcing with κ -Borel probability measures on the space of L -structures with a (possibly uncountable) infinite set X, focusing on those that are invariant under the action of the symmetric group Sym ( X ) . We demonstrate how any Sym ( X ) -invariant measure where X is countable can be uniquely extended to a Sym ( Y ) -invariant measure where Y is uncountable, and prove that forcing with such measures satisfies the countable chain condition. We also show that we can uniformly distinguish between these random generic structures and the Cohen generic structures that arise from forcing with a strong Fraïssé class: There is a κ -Borel set of low complexity that contains every Cohen generic structure that is not highly homogeneous but contains no random generic structure, implying that a structure that is not highly homogeneous cannot be both Cohen generic and random generic. Finally, we answer an open question of Kostana in the case of ω 1 , by establishing a connection between forcing with a strong Fraïssé class and Cohen forcing.
MIT Department
Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences
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DOI of Published Version
https://doi.org/10.1007/s11787-025-00394-2
