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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]