The australian philosopher colin leslie dean points out Mathematicians are
in deep crisis for 2 reasons
1) Skolems paradox
2) Godels incompleteness theorem is meaningless as he does not tell us what statements true
1) skolem discovered a paradox which makes set theory inconsistent
of which freankel and most mathematicians at the time saw
http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.
Quote:Neither have the books yet been closed on the antinomy, nor has
agreement on its significance and possible solution yet been reached." -
(Abraham Fraenkel)
and that
"most mathematicians followed fraenkels skepticiam
wiki states the paradox/contradiction thus
Skolem's paradox - Wikipedia, the free encyclopedia
Quote:contradiction, since the uncountable sets are subsets of the (countable) domain of the model.
[some will say only appears to be a contradiction but this comes from the belief that skolem gave a solution to the paradox - see below to see his solution destroys set theory and is not accepted]
note it says there is a contradiction in set theory ie that must make it inconsistent
note most mathematicians at the time including Abraham Fraenkel saw it as an antinomy
some argue that Skolem gave a solution to the paradox but this is not accepted as it destroys set theory
Quote: A widely held interpretation is that of Thoralf Skolem himself. He believed that LST showed a relativity in some of the fundamental concepts of set theory. Uncountable cardinalities in particular have no meaning apart from specific sets of axioms. A set may be uncountable within a certain formal system and countable when viewed from the standpoint of an unintended model of that system. This means that there simply are no sets whose cardinality is absolutely uncountable. For many, this view guts set theory, arithmetic, and analysis. It is also clearly incompatible with mathematical Platonism which holds that the real numbers exist, and are really uncountable, independently of what can be proved about them.
in regard to skolem relativism attempt at resolution - which is at
present is not accepted
Skolem himself admits his relativistic solution it destroys set theory
http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps
Quote:"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]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147
(John von Neumanns states
"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."
of which a few mathematician also agreed[/quote]
you have only two options
skolems paradox shows ZFC is inconsistent
or you accept skolems solution and thus ZFC is destroyed
now rather than solving the paradox before moving on with set theory
mathematicians just ignored it and used set theory for all sorts of
proofs
Now mathematicians are in crisis for there is now so much invested in
set theory that the skolem paradox threatens the very foundations of
mathematics
so some mathematician now try to argue away the paradox by saying it is
not a contradiction
but
skolems paradox want go away it is at present unable to be disproved
and modern maths is buried so much in rubbish for useing set theory they cant get out
2)
mathematician have so much invested in godels incompleteness theorem
much maths is reliant on it
but at the time godel wrote his theorem he had no idea of what truth was
as peter smith admit s the Cambridge expert on Godeladmits
but truth is central to his theorem
as peter smith kindly tellls us
http://assets.cambridge.org/97805218/57840/excerpt/9780521857840_excerpt.pdf
Quote:
Godel did is find a general method that enabled him to take any theory T
strong enough to capture a modest amount of basic arithmetic and
construct
a corresponding arithmetical sentence GT which encodes the claim 'The
sentenceGT itself is unprovable in theory T'. So G T is true if and only
if T can't prove it
If we can locate GT
, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
true-but-unprovable,
Gödel's incompleteness theorems - Wikipedia, the free encyclopedia...
without a notion of truth we dont know what makes those statements true
thus the theorem is meaningless
and modern mathematics is in deep **** for useing a meaningless theorem
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