Several contributors to this thread share Loader's confusion on a couple of points. For one, constructibility in set theory does have an isomorphism invariant meaning, as Mitchell showed in JPAA about 1975. Trivial changes make the definition work in any topos with natural number object. In fact, a good bit of work in set theory today is on finding isomorphism invariant consequences of V=L or related axioms (constructibility relative to some set). Elementhood is not essential. And Peter Freyd's question "Does any simple group appear as a zero of the Riemann Zeta function?" is not absolute nonsense in ZF or in category theory--it is only misleading as it makes a question of (arbitrarily definable) codings look like a question of complex analysis. In ZF the answer is: It depends which coding of groups and numbers you use--it is easy to make up codings where the answer is yes and just as easy to make them up where the answer is no. For what it is worth, the codings in common use all make the answer "no", since they make every group an ordered 3 or 4 tuple while they make a complex number an ordered pair of infinite sets. This is trivia. Now suppose we found some good relation representing groups as complex numbers, which was manageable to work with and for which we could show that some simple groups correspond to points which would falsify the Riemann hypothesis. Then mathematicians might well start asking whether some zero of the zeta function "is" a simple group--just as we now ask whether some given real number "is" rational (a question whose strict answer on most ZF codings is always "no"). And this question would be just as well expressed in category theory as in ZF. best regards, Colin