```Date: Jan 27, 2013 12:44 PM
Author: Jesse F. Hughes
Subject: Re: ZFC and God

WM <mueckenh@rz.fh-augsburg.de> writes:> 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.>> 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 "...".Are you suggesting that 0.777... is *both* an infinite and terminatingexpansion?Anyway, you haven't proved that there is a function   f:{1,...,k} -> {0,...,9}  as required by *your* definition of terminating decimal, so you havenot shown that 0.777... is a terminating decimal.> 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.Well, I don't understand why anyone would wish to show that.  But,regardless, this is beside the point.  I'm asking for a proof that0.777... is terminating according to the definition of terminatingthat you agreed to.Can you please show that proof?  Much thanks.-- "A signature block is meant for the author of a post. If the name ofsomeone other than the author of a post appears 'lastly in thepost'.[sic] It is taken to mean that such a person is the author."                        -- Archimedes Plutonium doesn't get .sigs
```