Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Finitely definable reals.
Posted:
Jan 16, 2013 2:32 AM
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In article
<0fa9e479-8702-4088-890e-8c1c70f4a4c1@v7g2000yqv.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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.
One must first establish that induction is allowed in ZF, but without
the axiom of infinity induction is NOT allowed in ZF.
>
> The induction procedure establishes that there is no actually infinite
> set by proving that the number of elements up to every natural is
> finite.
But it also shows that no finite set is maximal, the axiom of infinity
requires that for every finite set there is set with more members.
> 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.
That is not deriveable anywhere but in WMYTHEOLOGY
While for every n in |N there is an m in |N with m > n, it is only in
WMytheology that one can argue that this requires any m in |N greater
than all n in |N.
In standard mathematics, outside of WMytheology, WM's argument fails.
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