Re: Terminology of locally small categories without replacement

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 12/1/2010 2:00 PM, Colin McLarty wrote:
> These two [weak and strong notions of locally small] are not
  > equivalent in the absence of the axiom scheme of
> replacement.  There the second is much stronger, but it remains
> important.  Is there a good term for it?

Sure: "Locally small."  In the absence of Replacement it would make more
sense to call the weaker concept "weakly locally small" than the
stronger one "strongly locally small" since it is presumably the strong
one that is more often intended.  As you say, Replacement identifies the
concepts, and one then defines the common concept with whichever
definition is shorter or simpler, namely the weak one.

A downside of allowing multiple set theories is the proliferation of a
menagerie of definitions.  Considerations like the above can help manage
the menagerie, though the benefit of the menagerie in the first place
would seem to accrue more to logic than to mathematics.  The role of
logic in mathematics should be to understand the latter, not to
complicate it.

Vaughan


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