Skolem's paradox

In mathematical logic, specifically model theory, Skolem's paradox is a direct result of the (downward) Löwenheim-Skolem theorem, which states that every model of a sentence of a first-order language has an elementarily equivalent countable submodel.

Quotes

Zermelo at first declared the Skolem paradox a hoax. In 1937 he wrote a small note entitled "Relativism in Set Theory and the So-Called Theorem of Skolem") in which he gives a refutation of "Skolem's paradox", i.e. the fact that Zermelo-Fraenkel set theory --guaranteeing the existence of uncountably many sets-- has a countable model. Other authorities on set theory also found the result astounding.

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• At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known. (John von Neumann)
• Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached. (Abraham Fraenkel)
• I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique. (Skolem)