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Topic: How many real numbers are there?
Replies: 18   Last Post: Jul 9, 2006 3:00 PM

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Stephen Montgomery-Smith

Posts: 2,351
Registered: 12/6/04
Re: How many real numbers are there?
Posted: Jul 7, 2006 11:35 PM
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Nathan wrote:
> Mike Deeth wrote:

>>It's well-known that a countable union of finite sets is countable.
> Nope! You said you were working in ZF. This requires AC.

1. This isn't the flaw in his argument, and
2. The place where you need AC (or some weaker version thereof) is that
you first have to choose an enumeration of each set. But my feeling is
that if you go through his argument carefully, that you can make this
choice by construction.

No, the real reason his argument fails (and someone else has said this)
is that he has merely proved that the set of constructable real numbers
is countable. Since there exists a bona fide proof that the reals are
uncountable, he has in effect shown that there are real numbers which
are not contructable. In fact one can apply the diagonalization
argument directly and construct a non-constructable real number, showing
that the notion of constructable real number needs careful definition,
and cannot be defined in a completely canonical fashion.

Best Stephen

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