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 such that every non-Pseudoautistic category is isomorphic to a category .1

Proof

Suppose towards contradiction that is a universal category of categories, and be the full subcategory consisting of all categories which are not pseudoautistic. Then is a category of categories containing (up to isomorphism) and .

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

Now by Simpson's lemma, is not pseudoautistic, and by universality there exists such that , hence is not pseudoatustic and thus , so is autistic, a contradiction.

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Footnotes

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