Re: Is Set lfp, intuitionistically?

On Mon, 1 Oct 2001, Michael Abbott wrote:

Or, I hope this is the same question, is an elementary topos with a natural numbers object internally locally finitely presentable? Are there any references for this?

The answer is yes, but (like a great many such things) I don't think it is written down anywhere. Finite cardinals are internally finitely presentable (the proof of this is similar to the proof that they are internally projective, see 6.25 in "Topos Theory"), and the fact that every object is internally a filtered colimit of finite cardinals is implicit in the construction of the object classifier (cf. 6.32 in the same reference). Peter Johnstone