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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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Jonathan Hoyle

Posts: 754
Registered: 12/10/04
Re: How many real numbers are there?
Posted: Jul 8, 2006 12:44 AM
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> Nowhere in the constructions of the real numbers do we make any
> assumptions that we are in some exotic set theory with supernatural
> powers. This is not X-Men mathematics! The set T which I contructed
> in my original post satisfies all the axioms of ZF...

Up to this point you are correct.

> ...and therefore the contructions of the reals work fine therein.

This is where your fallacy lay. Just because all the sets so
contructed are in ZF does not imply the other way around. You seem to
be adding the following axiom:

NA (The Nathan's Axiom): All sets are finitely constructible.

Theorems of ZF + NA are not the same ones from ZF alone. From ZF + NA,
you can deduce that there are only countably many reals (since most
transcendental reals will not exist in ZF + NA). However, this
statement is false in ZFC.

There is a branch of study called Constructible Set Theory (V=L), which
is similar to what you have, except it restricts sets to be
constructible (but not necessarily finitely). In this universe, both
the Axiom of Choice (AC) and the Continuum Hypothesis (CH) become
theorems. In your ZF + NA, the AC remains a theorem, but CH fails
since there are only countably many sets in your theory. You may be
interested in checking this out:

In any case, ZF + NA is not ZFC, and even proponents of V=L would not
agree with NA.

Jonathan Hoyle

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