False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-05 (gpt-5.6-sol-codex-subscription)‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: $\aleph_\alpha$ is regular for every ordinal $\alpha$

Statement

FALSE. Every aleph is regular: $\operatorname{cf}(\aleph_\alpha) = \aleph_\alpha$ for every ordinal $\alpha$ (Cofinality $\operatorname{cf}(\alpha)$ , and regular and singular cardinals, The successor cardinal $\kappa^{+}$ , the alephs $\aleph_\alpha$ , the beths $\beth_\alpha$ , successor and limit cardinals, and the identifications $\aleph_0 = \omega$ and $\aleph_1 = \omega_1$ ).

The claim is plausible because the alephs a reader meets first are regular: $\aleph_0$ is regular in ZF, and assuming the Axiom of Choice every successor aleph is regular ( $\aleph_0$ is regular in ZF; assuming the Axiom of Choice every successor aleph $\aleph_{\alpha+1}$ is regular; $\operatorname{cf}(\aleph_\omega) = \aleph_0$ , so $\aleph_\omega$ is singular, and under choice it is the least singular infinite cardinal). It fails at the first aleph whose index is a limit ordinal, and the failure is a theorem of ZF requiring no choice principle at all.

Facts & Assumptions

Given: The alephs of The successor cardinal $\kappa^{+}$ , the alephs $\aleph_\alpha$ , the beths $\beth_\alpha$ , successor and limit cardinals, and the identifications $\aleph_0 = \omega$ and $\aleph_1 = \omega_1$ and the cofinality of Cofinality $\operatorname{cf}(\alpha)$ , and regular and singular cardinals.

[L1]

$\operatorname{cf}(\aleph_\omega) = \aleph_0$ , and $\aleph_0 < \aleph_\omega$ , so $\aleph_\omega$ is singular; this is a theorem of ZF (clause (c) of $\aleph_0$ is regular in ZF; assuming the Axiom of Choice every successor aleph $\aleph_{\alpha+1}$ is regular; $\operatorname{cf}(\aleph_\omega) = \aleph_0$ , so $\aleph_\omega$ is singular, and under choice it is the least singular infinite cardinal).

[L2]

An infinite cardinal $\kappa$ is regular when $\operatorname{cf}(\kappa) = \kappa$ , and singular when $\operatorname{cf}(\kappa) \ne \kappa$ (Cofinality $\operatorname{cf}(\alpha)$ , and regular and singular cardinals, Cardinal (initial ordinal) and cardinality).

[L3]

The operation $\alpha \mapsto \aleph_\alpha$ is defined at every ordinal and is strictly increasing (The clauses at $0$ , at a successor and at a limit determine exactly one operation $\alpha \mapsto \aleph_\alpha$ , in ZF, and — assuming the Axiom of Choice — exactly one operation $\alpha \mapsto \beth_\alpha$ ; each value is an infinite cardinal, each is strictly increasing and continuous at limits, and $\alpha \le \aleph_\alpha$ ).

[L4]

Ordinals satisfy trichotomy and $\alpha \notin \alpha$ , so $\alpha < \beta$ and $\alpha = \beta$ cannot both hold (Trichotomy and well-ordering of the ordinals, Basic closure properties of ordinals).

Refutation

technique · contradiction

1.1

Suppose, for contradiction, that $\operatorname{cf}(\aleph_\alpha) = \aleph_\alpha$ for every ordinal $\alpha$ .

assume-contra

1.2

By [L1] and [L3], $\operatorname{cf}(\aleph_\omega) = \aleph_0$ and $\aleph_0 < \aleph_\omega$ .

L1L3

2.1

Instantiating the assumption at $\alpha = \omega$ gives $\operatorname{cf}(\aleph_\omega) = \aleph_\omega$ , hence $\aleph_0 = \aleph_\omega$ by step 1.2, which [L4] forbids; so not every aleph is regular, and $\aleph_\omega$ is singular by [L2].

step 1.1step 1.2L2L4discharge-contradiction∎

Remarks

Which alephs the theorem does certify. $\aleph_0$ , in ZF; and every $\aleph_{\alpha+1}$ , assuming the Axiom of Choice ( $\aleph_0$ is regular in ZF; assuming the Axiom of Choice every successor aleph $\aleph_{\alpha+1}$ is regular; $\operatorname{cf}(\aleph_\omega) = \aleph_0$ , so $\aleph_\omega$ is singular, and under choice it is the least singular infinite cardinal). So the false claim is not wrong everywhere — it is wrong exactly where the index is a limit ordinal reached from below by a short family, and $\omega$ is the smallest such index.

Singularity here is about the index, not about the size. The cofinal family that witnesses $\operatorname{cf}(\aleph_\omega) = \aleph_0$ is $n \mapsto \aleph_n$ , indexed by $\omega$ because the subscript $\omega$ is a limit of an $\omega$ -sequence. Nothing about how large $\aleph_\omega$ is enters the argument, and nothing is chosen, which is why clause (c) of $\aleph_0$ is regular in ZF; assuming the Axiom of Choice every successor aleph $\aleph_{\alpha+1}$ is regular; $\operatorname{cf}(\aleph_\omega) = \aleph_0$ , so $\aleph_\omega$ is singular, and under choice it is the least singular infinite cardinal is choice free while clause (b) is not.

The claim is not repaired by assuming choice. Adding the Axiom of Choice certifies more alephs as regular, but it does not touch the witness above: the refutation is a theorem of ZF and remains one in ZFC.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 87 results over 28 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

UCL, Axiomatic Set Theory, Ch. 4: Cardinal Arithmetic (standard reference, not scraped)
Regular cardinal (Wikipedia) (standard reference, not scraped)
Cofinality (Wikipedia) (standard reference, not scraped)

FALSE: ℵα\aleph_\alphaℵα​ is regular for every ordinal α\alphaα