The Automorphism Threshold

Nontrivial automorphisms of P(ω)/Fin exist when you add fewer than ℵ_ω Cohen reals. The threshold is itself a structural boundary.

P(ω)/Fin is the Boolean algebra of subsets of natural numbers modulo finite sets. Its automorphisms rearrange infinite subsets in ways that are consistent up to finite error. Under the Continuum Hypothesis (CH), Shelah and Steprāns proved that nontrivial automorphisms exist — rearrangements that cannot be induced by any function on the natural numbers. But CH is a strong assumption. The question: how far from CH can you get and still have nontrivial automorphisms?

Brian and Dow (arXiv:2603.07214) prove that adding fewer than ℵ_ω Cohen reals to a model of CH preserves nontrivial automorphisms. The previous result (Shelah and Steprāns) was known only for ℵ₂ — adding one Cohen real. This extends it to ℵ₃, ℵ₄, all the way up to any cardinal below ℵ_ω. The threshold at ℵ_ω is not arbitrary — it is the first limit cardinal, where the inductive method used in the proof necessarily breaks.

The structural point: the existence of nontrivial automorphisms is controlled by the cardinal arithmetic of the forcing, not just its size. Adding reals makes the combinatorics of infinite subsets more complex, but below ℵ_ω the added complexity is not enough to destroy the automorphisms that CH provides. At ℵ_ω, the inductive construction collapses because the limit ordinal has no predecessor to anchor the next step.

The threshold between “nontrivial automorphisms exist” and “they might not” is a boundary in the hierarchy of infinities, not a boundary in set-theoretic axioms.

Brian and Dow, “Nontrivial automorphisms of P(ω)/Fin in Cohen models,” arXiv:2603.07214 (2026).