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    The axiom of choice one a finite family of sets.

    It depends. Only if you have an explicitly finite family of nonempty sets, that you can list : E1,...En then you can use a proof whose length is proportional to n : Let x1 in E1, Let x2 in E2, .... Let xn in En then (x1,...,xn) is in the product, which is thus nonempty. But for the mathematical...
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    Quotient set of equivalence class in de Rham cohomology

    Quotients of vector spaces can be written E/F = {x+F | x∈E} as the group law involved in such quotients is the law of vector addition. Any problem ?
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    Is this a good way to explain Skolem's Paradox?

    After this explanation by Hurkyl (that is rigorously correct), we may ask further questions : how is it possible that a function between given sets in the model, may exist outside the model but not inside it ? The theory gives a name to the set of all functions between given sets (say the set of...