The Skolem paradox destroys the incompleteness of ZFC

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The Skolem paradox challenges the consistency of Zermelo-Fraenkel set theory (ZFC), suggesting that the undecidability of ZFC relies on the assumption of its consistency. Colin Leslie Dean argues that the paradox undermines established proofs of ZFC's incompleteness, labeling them as invalid. The paradox arises from the Löwenheim-Skolem Theorem, which allows for a model of set theory that is countable yet contains uncountable sets, creating a contradiction. Critics express skepticism about Dean's claims, questioning his credibility and the legitimacy of his arguments. Overall, the discussion highlights ongoing debates regarding the foundations of set theory and its implications for mathematics.
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The Australian philosopher colin leslie dean argues that
The Skolem paradox destroys the incompleteness of ZFC

Crackpot link removed[/color]

The Skolem pardox shows ZFC is inconsistent
Undecidability of ZFC is based on the assumption that it is consistent
therefore
the presence of the Skolem paradox shows ZFC is not consistent
so all those proofs that show the incompleteness of ZFC are destroyed
undermined and complete rubbish
 
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from colin leslie dean

Crackpot link removed[/color]

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." – (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'." – (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)
 
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The Australian "philosopher" colin leslie dean seems to have been extremely drunk when he wrote this paper.
 
I suspect that anyone who publishes through something called the "gamahucher press" spends a fair amount of time drunk.

I also suspect, though not as surely, that "gamel" is "The Australian philosopher colin leslie dean" and runs that press.
 
WHAT DOES gamahucher MEAN i wonder
 
CLD is a crackpot, and you were banned once already for this.
 
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I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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