Re: The Skolem paradox destroys the incompleteness of ZFC

Skolem paradox does not show that ZFC is inconsistent.


sorry the Skolem paradox does show ZFC is inconsistent

http://en.wikipedia.org/wiki/Skolem's_paradox

The paradox is seen in Zermelo-Fraenkel set theory. One of the earliest
results, published by Georg Cantor in 1874, was the existence of
uncountable sets, such as the powerset of the natural numbers, the set of
real numbers, and the well-known Cantor set. These sets exist in any
Zermelo-Fraenkel universe, since their existence follows from the axioms.
Using the Löwenheim-Skolem Theorem, we can get a model of set theory
which only contains a countable number of objects. However, it must
contain the aforementioned uncountable sets, which appears to be a
contradiction.


"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." â?? (John von
Neumann)

"Skolem's work implies 'no categorical axiomatisation of set theory
(hence geometry, arithmetic [and any other theory with a set-theoretic
model]...) seems to exist at all'." â?? (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." â??
(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." â?? (Skolem)

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