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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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Anonym1723

Posts: 457
Registered: 12/12/04
Re: How many real numbers are there?
Posted: Jul 9, 2006 1:45 AM
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> 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.


I think that ZF + NA is inconsistent, since NA contradicts the Axiom of
Replacement. All those sets he created S_i for finite integer i should
be replaceable with any ordinal i, creating sets of infinite
constructibility.

Even in Nathan's own unstated axioms, his world is inconsistent with
ZF.

Note that this isn't a problem in the Constructible universe of V=L,
since Nathan's axiom does not hold, and there are still uncountably
many reals.




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