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Topic: Finitely definable reals.
Replies: 52   Last Post: Jan 18, 2013 2:37 PM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Finitely definable reals.
Posted: Jan 16, 2013 1:49 AM

On 15 Jan., 21:06, Virgil <vir...@ligriv.com> wrote:

> The induction procedure exhausts both the first natural  and also the
> successor of every natural, so what naturals does it fail to exhaust?

If the induction procedure was sufficient to establish an actually
infinite set, ZF would not need the axiom of infinity.

The induction procedure establishes that there is no actually infinite
set by proving that the number of elements up to every natural is
finite.
As this holds always, the naturals are potentially infinite but we
never gather a number of naturals that is larger than any natural. We
get it only by reversion of quantification. A simple mistake:

For every n there is m > n
has been confused to
There is an m larger than every n.

Ridiculous.

Regards, WM