Grothendieck

dimbulb: The notion of "logic" in the standard foundations of mathematics is a precursor to the notion of "set theory". Logic can be done thinking only about writing symbols on a piece of paper in a coherent way (i.e. following metamathematical rules).

In logic, you have things called "terms" and "statements", without any "surrounding set theory". Thus the notation (x) just means "for all x", not "for all x in a given set", cf. the fact that there is no set mentioned. The dot means "and", its as simple as that.

Part of the mathematical discipline of "model theory" is finding models for logic in set theory. This means replacing statements like "Px" by explicit statements about explicit sets. "For all x" gets replaced by a statement about all sets in a given "big set" which we call universes. We need these universes to avoid set theoretical paradoxes, the best known of which is the set containing all sets which do not have themselves as elements. Does that set contain itself as an element?

Nobody really wants to know about these things though, except professionals, do they? I agree with David2000 on that one...

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