Virgil
Posts:
4,486
Registered:
1/6/11
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Re: ZFC and God
Posted:
Jan 27, 2013 4:14 PM
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In article
<6cfca275-f73b-4810-80c5-3b24ee884692@m12g2000yqp.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 27 Jan., 18:18, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > WM <mueck...@rz.fh-augsburg.de> writes:
> > > On 27 Jan., 15:49, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> >
> > >> >> That is, for each i in N, the i'th digit of 0.777... is defined and is
> > >> >> 7.
> >
> > >> > And do you have problems to find this confirmed as possible in the
> > >> > complete set of terminating decimals? Any digit or index missing?
> >
> > >> I've no idea what you mean when you ask whether I can "find this
> > >> confirmed as possible". But, for each i in N, the i'th digit of
> > >> 0.777... is defined and equals 7. Is there anything more I need to
> > >> know in order to claim that it is a non-terminating decimal?
> >
> > > You need to know whether this n is an element of a finite initial
> > > segment of {1, 2, 3, ..., n, n+1, n+2, ..., n^n}.
> >
> > [SNIP]
> >
> > Sorry, let's focus on the question at hand. I fear that your response
> > diverts from the issue I want clarified. (Once again, you've
> > inadvertently snipped my primary question.)
> >
> > By definition,
> >
> > 0.777... = sum_i=1^oo 7*10^-1.
> >
> > You claim that 0.777... has a terminating decimal representation
> > (right?).
>
> We are working in the domain of terminating decimals. Unless you can
> find an index of a digit of 0.777... that does not belong to a finite
> initial segment {1, 2, ..., n} of the natural numbers, 0.777...
> belongs to that domain.
>
> >
> > You accept the following definition:
> >
> > Let x be a real number in [0,1]. We say that x has a terminating
> > decimal representation iff there is a natural number k and a
> > function f:{1,...,k} -> {0,...,9} such that
> >
> > x = sum_i=1^k f(i) * 10^-i.
> >
> > Therefore, I request a proof that there is a function
> >
> > f:{1,...,k} -> {0,...,9}
> >
> > such that
> >
> > sum_i=1^k f(i)*10^-i = sum_i=1^oo 7*10^-i.
> >
> > Unless you can prove that there is such a function, we must conclude
> > you have no proof that 0.777... is terminating.
>
> Unless you can prove that there is a digit 7_i with an i that does not
> belong to a finite initial segment of the natural numbers, I see no
> necessity to prove anything. We must conclude you have no proof that
> 0.777... is longer than every terminating sequence, namely actually
> infinity.
The union of any set of sets is itself a set so the union of the set of
all fisons is a set and it contains for every fison a natural number
not in it is, so that union cannot be itself a fison.
>
> But here is the proof that we can work in the domain of terminating
> decimals including 0.777...:
>
> 0.7 is terminating.
> if 0.777...777 with n digits is terminating, then also 0.777...7777
> with n+1 digits is terminating. Therefore there is no upper limit for
> the number of digits in a terminating decimal. This fact is usually
> denoted by "infinite" and abbreviated by "...".
But you still have no proof that we MUST work with only terminating
decimals, and until you do we won't.
>
> Note, there is another meaning of infinite, namely "actually
> infinite". Those who adhere to that notion *in mathematics* should
> show that it differs from "potentially infinite" *in mathematics*,
> i.e., expressible by digits.
We find that the union of all fisons is a set containing all and only
natural numbers, which set we denote by |N.
Our definition of a set being infinite is that there exist an injection
from |N to that set.
We do not recognize any distinction between what WM calls "potential
infiniteness" and "Actual infiniteness".
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