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False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11
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Every category is locally small

Statement

FALSE. Every category is locally small.

Facts & Assumptions

Given: The definable class 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 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 O with object ∗ and Hom⁡O(∗,∗)=Ord. Take 0 as the identity and define β∘α=max⁡{α,β}.

givenL2L3
2.1

Maximum is associative, and max⁡{0,α}=α=max⁡{α,0} for every ordinal α. 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 Ord, which is not a set by [L3]. Hence O is not locally small by [L1].

step 1.1L1L3
3.1

The category 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 · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources