Re: Mapping of Reals to Naturals [What is the flaw in the proof?]
- From: Virgil <virgil@xxxxxxxxxxx>
- Date: Wed, 18 Apr 2007 23:27:53 -0600
In sci.math, Amitabh
<amitabh123@xxxxxxxxx>
wrote
on 16 Apr 2007 05:38:07 -0700
<1176727087.849639.274080@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>:
Cantor proved that the set of reals cannot be mapped to the set of
integers (using a one-to-one mapping)
I seem to have proved that every element in the power set of integers
can be uniquely mapped to another integer. Is there any flaw in my
argument.
Let N be the set of naturals. We consider P, the set of all primes. We
know that |P| = |N|.
Now consider the set 2^P.
Each element of 2^P can be uniquely mapped to an element of N as
follows:
each element of 2^P is a set of distinct prime numbers. If we multiply
all the primes in any such set, we get a unique natural number. Thus
we
can map each element of 2^P to N, contradicting Cantor's
diagonalization
proof that 2^N cannot be mapped to N.
Notice that some (actually most) of the elements of 2^P contain
infinitely many primes so that the alleged product of those infinitely
many primes cannot be a natural number, and your proof crashes.
.
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- Mapping of Reals to Naturals [What is the flaw in the proof?]
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