Date: Jan 27, 2013 4:14 PM Author: Virgil Subject: Re: ZFC and God 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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