Mathematicians are in crisis for 2 reasons

  1. Philosophy Forum
  2. » Logic
  3. » Mathematicians are in crisis for 2 reasons

Get Email Updates Email this Topic Print this Page

pam69ur
 
Reply Tue 29 Apr, 2008 12:58 am
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

Quote:
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...

Quote:
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
 
Arjen
 
Reply Tue 29 Apr, 2008 10:26 am
@pam69ur,
I am wondering if you mean that maths is in trouble because formulae are based on the incompleteness theorem and that the formula is incorrect in itself or that the theory the incompleteness theorem presents (especially in combination with skolems paradox which helps point just that out) makes math "obsolete" in a way?

I think both are incorrect by the way. If there is another thing you ment by this I would like to know what exactly. To me both the paradox and the theorem made everything a lot clearer you see.
 
pam69ur
 
Reply Tue 29 Apr, 2008 02:39 pm
@pam69ur,
Quote:
I am wondering if you mean that maths is in trouble because formulae are [1]based on the incompleteness theorem and that the formula is incorrect in itself or [2] that the theory the incompleteness theorem presents (especially in combination with skolems paradox which helps point just that out) makes math "obsolete" in a way?


i am saying 1
because
The incompleteness theorem is meaningless it is worthless as it does not tell us what true statements are

wiki gives the theorem as

Gödel's incompleteness theorems - Wikipedia, the free encyclopedia
Quote:
, perhaps the single most celebrated result in mathematical logic, states that:
For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.


now godel had no idea what truth was
so he cant tell us what makes a ststement true
thus his theorem is meaningless /worthless

skolems paradox means set theory ie ZFC is inconsistent thus everyu thing mathematicians have done or proved with ZFC is worthless
or
if you accepts skolems solution -which is not accepted-then ZFC is destroyed
 
Aristoddler
 
Reply Tue 29 Apr, 2008 09:03 pm
@pam69ur,
Wasn't Godel's theorem debunked? Or was that Kant?
 
pam69ur
 
Reply Tue 29 Apr, 2008 09:15 pm
@pam69ur,
Quote:
Wasn't Godel's theorem debunked? Or was that Kant?


well colin leslie dean has shown godels theorem is invalid as it
1) uses the invalid axiom of reducibility
2) he constructs impredicative statements which texts books on logic say are invalid

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
Quote:
GODEL'S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS GODEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS
 
Arjen
 
Reply Wed 30 Apr, 2008 06:01 am
@pam69ur,
 
 

 
  1. Philosophy Forum
  2. » Logic
  3. » Mathematicians are in crisis for 2 reasons
Copyright © 2021 MadLab, LLC :: Terms of Service :: Privacy Policy :: Page generated in 0.02 seconds on 09/25/2021 at 07:02:08