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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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Mike Deeth

Posts: 25
Registered: 12/13/04
Re: How many real numbers are there?
Posted: Jul 7, 2006 11:06 PM
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Eric Schmidt wrote:
> (snip)
> You have not proved that every set (or even every real number) is in
> some S_i. In fact, you yourself construct a set not in any S_i, namely
> T. Therefore you have not found any flaw in set theory.

Of course my T does not contain every set, that would be really silly.
However, the elements of my set T satisfy the ZF axioms. They do this
by very construction, and you cannot argue about that! For example, if
x and y are in my set T, then so is x union y, {x,y}, etc.

The construction of the real numbers does not depend on the specific
details of the set theory. *** All it requires is that ZF axioms hold,
and they DO hold for elements of my set T *** Therefore, we can
construct the "reals" (ie, a complete ordered field) using nothing but
the elements of my set T. My set T has countably many elements.
THEREFORE the cardinality of the reals is countable!!

If you want to dispute this, please consider the following string of
statements and tell me which one is false. Just one will suffice.

1. The construction of the reals (ie, a complete ordered field) does
not depend on particular details of one's set theory, only on the fact
that the ZF axioms hold

2. ZF axioms hold for elements of my set T

3. We can construct, in the usual way, the "reals" (ie, a complete
ordered field), in such a way that each "real" is an element of my set

4. My set T has countable cardinality

5. THEREFORE, the "reals" (ie, one particular complete ordered field,
namely the one in statement 3) have countable cardinality.

Note that in making the deduction in step 5 we use the lemma "A subset
of a countable set is countable". As some point out, this might take a
little more power than ZF. But this is fine! My set T was constructed
in ZFC! The *elements* of my set T may or may not satisfy a choice
axiom. The complete ordered field constructed in statement 3 consists
of elements of T, and each element of T is a set in ZFC, and therefore
in the original ZFC I started out with, the complete ordered field of
statement 3 is a subset of the countable set T, and thus is countable.

The *very most* the evil Cantorians can salvage at this point would be
a statement like, "the assertion that the reals are uncountable, like
the axiom of choice or continuum hypothesis, is an independent
assertion and can't be proved or disproved".

I hope you're having a wonderful day,
Nathan the Great
Age 11 :-)

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