False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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.

Every category is locally small

Statement

FALSE. Every category is locally small.

Facts & Assumptions

Given: The definable class $\mathrm{Ord}$ of all ordinals, interpreted under the category-size convention in Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{CAT}$ is not formed.

[L1]

A category is locally small exactly when every hom-collection is a set (Small, locally small, and large categories).

[L2]

Ordinals are linearly ordered by membership, so any two have a maximum (Ordinal (von Neumann), Trichotomy and well-ordering of the ordinals).

[L3]

There is no set of all ordinals (Burali-Forti: there is no set of all ordinals).

Refutation

technique · direct

1.1

Define a one-object category $\mathcal O$ with object $*$ and $\operatorname{Hom}_{\mathcal O}(*,*)=\mathrm{Ord}$ . Take $0$ as the identity and define $\beta\circ\alpha=\max\{\alpha,\beta\}$ .

givenL2L3

2.1

Maximum is associative, and $\max\{0,\alpha\}=\alpha=\max\{\alpha,0\}$ for every ordinal $\alpha$ . Thus the data in step 1.1 satisfy the category axioms under the given convention.

step 1.1givenL2

2.2

Its sole hom-collection is $\mathrm{Ord}$ , which is not a set by [L3]. Hence $\mathcal O$ is not locally small by [L1].

step 1.1L1L3

3.1

The category $\mathcal O$ of step 2.1 refutes the assertion that every category is locally small.

step 2.1step 2.2∎

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: 26 results over 14 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

Emily Riehl, Category Theory in Context, sections 1.1 and 1.2 (standard reference, not scraped)