Category theory MOC

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 ∈𝖣.¹

Proof

Suppose towards contradiction that 𝖢 is a universal category of categories, and 𝖣 ⊆𝖢 be the full subcategory consisting of all categories which are not pseudoautistic. Then 𝖣 is a category of categories containing (up to isomorphism) Interval category and [[Ordinal category|𝟥]].

Suppose, again towards contradiction, that 𝖣 is autistic. Then there exists some category ˜𝖣 ∈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 ∈𝖢 such that 𝖣 ≅˜𝖣, hence ˜𝖣 is not pseudoatustic and thus ˜𝖣 ∈𝖣, so 𝖣 is autistic, a contradiction.

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Footnotes

1. 1999. FOM: Russell paradox for naive category theory ↩