Date: Jan 22, 2013 4:39 PM
Author: Virgil
Subject: Re: ZFC and God

In article 
WM <> wrote:

> On 22 Jan., 17:48, "Jesse F. Hughes" <> wrote:
> > WM <> writes:
> > > On 22 Jan., 15:49, "Jesse F. Hughes" <> wrote:
> >
> > > FIS: Finite Initial Segment
> > > FISON: Finite Initial Segment Of Naturals (or indices)

> >
> > >> Well, it's not a "union" in the usual sense, but let's let it pass.
> >
> > > It is a union in that sense that every FISON {1, 2, ..., n+1} added
> > > contains all smaler FISONs {1, 2, ..., n}.

> >
> > We were speaking about the real number d, not the set N.

> The set N is required to index the digits of the decimal fractions of
> the real numbers.

But in WMytheology, there is no such thing as a set |N, since WM does
not allow non-finite sets and any |N is provably not finite.
> >
> > > This does never change. In particular the set is always finite. If
> > > you add with always doubling frequence, you can add all FISONs in
> > > finite time - given that an "all" is meaningfull here. But at the
> > > end, you think, we cannot follow so quickly, and abracadabra we get
> > > something larger than every FISON? Not in mathematics!

> >
> > Look, you need to offer an actual proof that
> >
> >   U_n=1^oo {1,...,n} is finite.
> >
> > I'll give you a hint, by showing you the proof that
> >
> >   U_n=1^oo {1,...,n} is infinite.

> Unfortunately you don't seem to understand what infinite means, in
> particular that it has two different meanings.

To WM it seems to have infinitely many different meanings dependent is
some inexplicable way on what WM wants it to mean.
> >
> > Just so you know what a proof looks like.
> >
> > Here's the proof.  First, let me be clear what I mean by infinite.  I
> > mean that there is no natural k such that |U_n=1^oo {1,...,n}| = k.

> That is potential infinity. That proof is not necessary, because the
> set is obviously potentially infinite. No, you shoudl give a proof,
> that there is a larger k than all finite k.

So WM's definition of an infinite set is one which is larger than itself.
> >
> > Hence, by definition, U_n=1^oo {1,...,n} is infinite.

> Correct, there is no finite threshold.

Either that union of sets represents a set, which it does in all
standard set theories, or an illusion, which is what it represents in
> >
> > Now, I'm sure you're familiar with that argument.  I'd like to see
> > your claim to the contrary proved with just as much detail and I'd
> > like for you to be willing to answer any questions I raise about any
> > steps I don't understand.
> >
> > Until then, I simply cannot see your reasoning that the union is
> > finite.

> You have not proved that the cardinal number of the union surpasses
> all natural numbers. And even if you had done so, it would be useless,
> because the union of FISONs cannot surpass itself.

But the cardinal of the set of all naturals (or union of all FISONS) is
easily shown to exceed the cardinality of any natural,
and the union of all FISONs is easily shown to exceed each FISON.

So that in WM's world, it is possible to have f(x) for every x but not
for all x.
> >
> >

> > > From this "for every n" you claim "for all". And that is wrong for
> > > infinite sets.

> >
> > I have no idea what difference between "for every" and "for all" you
> > have in mind, but feel free to read the above as "for every n".

> I know that. But as I already mentioned, actual infinity requres "for
> all n".

How does it not rquire it with "for every n"?

Perhaps it is merely a quirk of German-Engish differences but in English
mathematics one cannot have "for every" without having "for all".