FOM: Re: Russell paradox for naive category theory

Todd Wilson twilson at
Tue May 11 18:34:41 EDT 1999

Stephen G Simpson 11 May 1999 15:15:22 (responding to Carsten Butz):
>  > The Russell paradox for naive category theory you try to sell here
>  > as new is at least 30 years old. ...
> You seem to be saying that my version of the Russell paradox for naive
> category theory (details in my posting of 11 May 1999 01:11:25) has
> been known for a long time.  I feel that there is some reason to doubt
> what you are saying (see below).  Therefore, I would appreciate a
> reference, even if the reference is only an informal allusion in some
> published paper, conference proceedings, newsgroup, oral history,
> report of an informal conversation, folklore, or whatever.

Here's one referece for this well-known piece of folklore (there are
no doubt others):  it appears as an elementary exercise (3L, p. 37),
in the book "Abstract and Concrete Categories" by Adamek, Herrlich,
and Strecker: 

    3L.  Quasicategories as Objects.  Show that one cannot form the
    "quasicategory of all quasicategories".  [Hint: Russell's paradox
    appears again.]

The context is a discussion on categories of categories.  In Chapter 2
of this book (titled "Foundations"), a distinction is made between
sets, classes, and conglomerates, essentially adding one more level of
"largeness" to the usual one.  These size distinctions lead to a
corresponding distinction between small categories, large categories,
and quasicategories.

Todd Wilson
Computer Science Department
California State University, Fresno

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