Russell's paradox for categories

One formulation of Russell's paradox for categories is about naïve category theory, i.e. category theory without a choice of foundations. It states that there cannot exist a Russellian category of categories $𝖣$ such that every non-Pseudoautistic category $𝖠$ is isomorphic to a category $˜ 𝖠 \in 𝖣$ .¹

Proof

Suppose towards contradiction that $𝖢$ is a universal category of categories, and $𝖣 \subseteq 𝖢$ be the full subcategory consisting of all categories which are not pseudoautistic. Then $𝖣$ is a category of categories containing (up to isomorphism) $2 ――$ and $𝟥$ .

Suppose, again towards contradiction, that $𝖣$ is autistic. Then there exists some category $˜ 𝖣 \in 𝖣$ such that $˜ 𝖣 ≅ 𝖣$ , so $˜ 𝖣$ is pseudoautistic and thus cannot be in $𝖣$ , a contradiction. Thus $𝖣$ is not autistic.

Now by Simpson's lemma, $𝖣$ is not pseudoautistic, and by universality there exists $˜ 𝖣 \in 𝖢$ such that $𝖣 ≅ ˜ 𝖣$ , hence $˜ 𝖣$ is not pseudoatustic and thus $˜ 𝖣 \in 𝖣$ , so $𝖣$ is autistic, a contradiction.

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1999. FOM: Russell paradox for naive category theory ↩

Russell's paradox for categories

Footnotes