Easton's theorem for the tree property below aleph_omega
Pith reviewed 2026-05-25 00:39 UTC · model grok-4.3
The pith
The tree property at every ℵ_n (1 < n < ω) is consistent with an arbitrary continuum function below ℵ_ω as long as 2^{ℵ_n} > ℵ_{n+1} for each n.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal ℵ_n, 1 < n <ω, is consistent with an arbitrary continuum function below ℵ_ω which satisfies 2^{ℵ_n} > ℵ_{n+1}, n<ω. Thus the tree property has no provable effect on the continuum function below ℵ_ω except for the restriction that the tree property at κ^{++} implies 2^κ>κ^+ for every infinite κ.
What carries the argument
A forcing construction over a ground model with infinitely many supercompact cardinals that preserves the tree property at the ℵ_n while controlling the continuum function via Easton-like forcing.
If this is right
- It is consistent to have the tree property at ℵ_2 while setting 2^{ℵ_1} to any value larger than ℵ_2.
- The continuum function below ℵ_ω can violate GCH at every ℵ_n while the tree property still holds at those cardinals.
- The tree property at κ^{++} imposes no restrictions on the continuum function beyond 2^κ > κ^+.
- Easton's theorem on the continuum function can be extended to include the tree property at the relevant cardinals.
Where Pith is reading between the lines
- The same large cardinal starting point might allow the tree property to coexist with other properties at ℵ_ω itself.
- The construction could be modified to control additional cardinal arithmetic features below ℵ_ω.
- Results of this type may generalize to sequences of cardinals of higher cofinality using similar supercompact assumptions.
Load-bearing premise
The ground model contains infinitely many supercompact cardinals.
What would settle it
A ZFC proof that the tree property at ℵ_3 implies 2^{ℵ_2} equals some specific value forbidden by the arbitrary functions satisfying only the inequality 2^{ℵ_2} > ℵ_3.
read the original abstract
Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal $\aleph_n$, $1 < n <\omega$, is consistent with an arbitrary continuum function below $\aleph_\omega$ which satisfies $2^{\aleph_n} > \aleph_{n+1}$, $n<\omega$. Thus the tree property has no provable effect on the continuum function below $\aleph_\omega$ except for the restriction that the tree property at $\kappa^{++}$ implies $2^\kappa>\kappa^+$ for every infinite $\kappa$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper establishes a consistency result in set theory: starting from a model with infinitely many supercompact cardinals, there is a forcing extension in which the tree property holds at every ℵ_n for 1 < n < ω, while the continuum function below ℵ_ω realizes any prescribed values satisfying the necessary inequalities 2^{ℵ_n} > ℵ_{n+1} for each n < ω. This is achieved via an Easton-style product or iteration that interleaves continuum forcing with steps preserving the relevant Aronszajn trees and supercompact embeddings.
Significance. If the result holds, it shows that the tree property at successors of regulars below ℵ_ω imposes no further restrictions on the continuum function beyond the classical 2^κ > κ^+ requirement at each relevant level. The construction supplies a direct analogue of Easton's theorem in the presence of the tree property, using the supercompact cardinals to obtain the necessary embeddings for preservation at each successive ℵ_n. The argument is a standard large-cardinal consistency proof and does not rely on ad-hoc parameters or self-referential definitions.
minor comments (3)
- §2, definition of the Easton product: clarify whether the support is Easton or full support at limit stages below ℵ_ω, as this affects the preservation of the tree property at ℵ_n for n ≥ 3.
- Theorem 1.1: the statement of the continuum function condition should explicitly note that it is only required for n < ω and that no condition is imposed at ℵ_ω itself.
- §4.2, the lifting argument for the supercompact embedding: add a sentence explaining why the generic filter for the tail forcing does not destroy the tree property at the image of ℵ_n.
Simulated Author's Rebuttal
We thank the referee for the positive summary of our result and the recommendation of minor revision. No specific major comments appear in the report.
Circularity Check
No significant circularity; standard consistency proof from external large-cardinal assumption
full rationale
The paper is a consistency theorem: it starts from a ground model with infinitely many supercompact cardinals (an external assumption) and constructs a forcing extension realizing the tree property at each ℵ_n (n≥2) together with an arbitrary Easton continuum function below ℵ_ω obeying the known ZFC inequalities. The derivation relies on standard forcing techniques, product/iteration arguments, and preservation lemmas that use the supercompact embeddings; none of these steps are self-definitional, fitted-input predictions, or reductions to self-citations. The central claim therefore remains independent of its own outputs and receives no circularity penalty.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Existence of infinitely many supercompact cardinals
Reference graph
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