A definition is called 'impredicative' iff it defines an object x by
reference to a set of which x is a member.

A proof is impredicative iff it involves an impredicative definition.

Cantor's theorem says that for all sets S there is no one-one function
from S to P(S), the set of all subsets of S. Cantor's diagonal proof
is a reductio that goes as follows:

Suppose there is a one-one function f:S -> P(S). Consider the set of
all members x of S such that ~xef(x). Call it D. D has to be in the
range of f; so there is a member d of S such that f(d)=D. Then deD <->
~deD.Q.E.D.

The definition of D involves f, which involves P(S), of which D is a
member if D exists. The keystone of the proof is that if f exists,
then D exists. But D seems to be impredicatively defined. If we ban
impredicative definitions, the proof fails.

Is that right? No predicativist accepts the diagonal proof? Are there
nuances as to what impredicative proofs are to be accepted according
to different predicativist stances?

Thanks

Tues, Feb 19 2008 9:54 pm was the last time I remember posting to this
book, but since my recent
appearance in a South Dakota newspaper that asks the question: Is
Archimedes Plutonium a genius.
Well, that answer is easy to answer.

I wrote a book correcting about 30 professors of mathematics who each
wrote a book wherein they
gave their own proof of the Euclid Infinitude of Primes and G. H.
Hardy was amoung that group
and there were other prominent mathematicians in that group. And Hardy
is considered a genius
but if Hardy could not deliver a valid Euclid Infinitude of Primes
proof and it took Archimedes Plutonium
to show where those 30 professors of math made a big mistake. And if
we consider Hardy
a genius, then we have to say that Archimedes Plutonium must be a
genius.

Tues, Feb 19 2008 9:54 pm was the last time I remember posting to this
book, but since my recent
appearance in a South Dakota newspaper that asks the question: Is
Archimedes Plutonium a genius.
Well, that answer is easy to answer.

I wrote a book correcting about 30 professors of mathematics who each
wrote a book wherein they
gave their own proof of the Euclid Infinitude of Primes and G. H.
Hardy was amoung that group
and there were other prominent mathematicians in that group. And Hardy
is considered a genius
but if Hardy could not deliver a valid Euclid Infinitude of Primes
proof and it took Archimedes Plutonium
to show where those 30 professors of math made a big mistake. And if
we consider Hardy
a genius, then we have to say that Archimedes Plutonium must be a
genius.

ICDM '08: The 8th IEEE International Conference on Data Mining
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hai,

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