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Science Forum Index » Logic Forum » the separation axiom is invalid therefore russells...
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| elsiemelsi... |
 Posted: Sun May 18, 2008 11:20 pm |
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The australian philospher colin leslie dean points out the separation
axiom is impredicative -
zermelo introduced it to outlaw the
russell paradox which showed naive set theory to be inconsistent- thus
russells paradox stands and set theory is inconsistent
because
impredicative statements are invalid according to text books on logic
poicare russell et al therefore the separation axiom is invalid and thus
russells paradox stands and set theory is inconsistent
http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
3. Axiom schema of specification (also called the axiom schema of
separation or of restricted comprehension): If z is a set, and \phi\! is
any property which may characterize the elements x of z, then there is a
subset y of z containing those x in z which satisfy the property. The
"restriction" to z is necessary to avoid Russell's paradox and its
variant
poincare and russell argued that impredicative statements led to paradox
in mathenmatics
now
the separation axiom of ZFC is impredicative
solomon feferman
http://math.stanford.edu/~feferman/papers/predicativity.pdf
"in ZF the fundamental source of impredicativity is the seperation axiom
which asserts that for each well formed function p(x)of the language ZF
the existence of the set x : x } a ^ p(x) for any set a Since the
formular
p may contain quantifiers ranging over the supposed "totality" of all the
sets this is impredicativity according to the VCP
this impredicativity is given teeth by the axiom of infinity "
now Adding to ZF either the axiom of choice (AC) or a statement
equivalent
thereto, yields ZFC.
thus an axiom which was invented ad hoc to outlaw a paradox in naive set
theory is it self impredicative and thus invalid thus russells paradox
stands and set theory is inconsistent
--
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| george... |
| Posted: Tue May 20, 2008 10:20 am |
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On May 19, 12:20 am, "elsiemelsi" <cyprin... at (no spam) nosam.yahoo.com> wrote:
Quote: The australian philospher colin leslie dean points out the separation
axiom is impredicative -
Neither you nor Dean can "point out" this fact, at least not to US:
WE ALREADY knew that the separation axiom is impredicative.
Quote: zermelo introduced it to outlaw the
russell paradox which showed naive set theory to be inconsistent-
True.
Quote: thus russells paradox stands
FALSE.
Introducing the separation axiom is not the ONLY thing that was done
here. Before that was done, we had to DELETE the OLD comprehension
axiom, the one that said (ALSO impredicatively) that you could make
a set out of ANY "concept" or "definite property". Since THAT is GONE
from ZF, the PRESENCE of anything (the separation axiom or anything
else) is just irrelevant. Russell's paradox DOES NOT occur in ZFC.
Neither does any other paradox, as far as YOU know.
What we call "Skolem's Paradox" IS NOT the KIND of paradox that
is inconsistency.
and set theory is inconsistent
Quote: because
impredicative statements are invalid according to text books on logic
How the fuck would YOU know??? It's NOT like YOU've ever READ one!
Quote: poicare russell et al therefore the separation axiom is invalid
Well, it is impredicative, but that is hardly the same thing as
"invalid".
Quote: and thus russells paradox stands
That IS NOT HOW "stands" WORKS around here.
There simply IS NO Russell's paradox TO stand or fall in ZF.
Russell's Paradox is in NAIVE set theory. With separation OR WITHOUT,
in Z, the paradox simply never comes into play.
Yes, ZF (where the relevant axiom is Replacement) is impredicative,
but that simply is not relevant to any question of consistency.
Outlawing impredicativity is one way to protect oneself from vicious
circles, but it is not the ONLY way. |
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