El 1/6/18 a las 13:21, streicher@mathematik.tu-darmstadt.de escribi?:

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Ha ha, you are right!?? G&V didn't make a mistake after all.?? Let me summarize.

Of course, I fully agree with your summary!

What a relief that we have converged at an agreement!

Thomas

It was a very good thing the problem in this thread is solved before the moderator stop the discussion, and with vert good reason. Questions on foundations attract a lot of outsiders, and we already have received a post where the problem was related to Ludwig Boltzman?s formulation of statistical mechanical entropy !!! I ask the moderator if he let me pass just one more post: ================ I was wrong when I wrote in a previous mail ?We are also outside U in the first case since a functional relation is a subset of C x U, and as such there is no way to prove it is a U-set, only in bijection with a U-set when C is a U-set.? As P Levy pointed out, Proposition 7 in Bourbaki's Appendix to SGA4. precisely shows that if C is a U-set, then any subset of C x U (note that C x U \subset U in this case) in bijection with a U-set also is a U-set. I note now that in an even earlier mail I said: ?However, Universes are closed in the sense: 2) X subset U, X ~ Y, Y belongs U? ===>? X belongs U (this is true because X is the union of its singletons which are in U) ? which is precisely Proposition 7 !! But even myself ignored this afterwards. Until P. Levy quoted Proposition 7, solve the question we were dealing in this thread, and wrote an excellent summary to explain this. ================== best Eduardo [For admin and other information see: http://www.mta.ca/~cat-dist/ ]