Date: Jan 25, 2013 2:52 AM Author: mueckenh@rz.fh-augsburg.de Subject: Re: ZFC and God On 25 Jan., 01:39, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> WM <mueck...@rz.fh-augsburg.de> writes:

> > On 24 Jan., 14:16, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> >> WM <mueck...@rz.fh-augsburg.de> writes:

> >> > You will have have recognized that here the diagonal argument is

> >> > applied. It is obvious that up to every line = column the list is a

> >> > square.

>

> >> It is clear that, for all j, d(j) != t_j(j) and hence d != t_j. If

> >> that's what you mean by the diagonal argument, great!

>

> >> Once again, however, you say something that has no clear meaning to

> >> me. Can you clarify "It is obvious that up to every line = column the

> >> list is a square?" I've no clue what it means.

>

> > Then ponder a while about the following sequence

>

> > d

>

> > d1

> > 2d

>

> > d11

> > 2d2

> > 33d

>

> > and so on. In every square there are as many d's as lines. The same

> > could be shown for the columns.

>

> Yes, in this sequence of three squares, what you say is true.

Is there a first square where my observation would fail?

>

> But none of this is relevant, because we've explicitly defined the

> anti-diagonal d and it is a triviality to see that it is an infinite

> sequence of non-zero and non-nine digits. And this fact really has

> nothing at all to do with limits of sequences of squares. It is all

> perfectly explicit.

Here you again intermingle potential and actual. We are restricted to

the domain of terminating decimals. If you cannot understand that,

perhaps a formal argument may help: Assume that we are restricted to

the well-defined set of terminating decimals. If you see any evidence

that we should leave that domain, say "stop!". But only if you are

sure.

>

> Do you agree that (by presumption) t_i is defined for every i in N?

Of course! Why not? Isn't every i in N finite?

>

> I don't want to imagine what you are thinking, because I will risk

> getting it wrong. I'd prefer that you explicitly give an argument in

> ZF so that we can determine whether it is valid or not.

In ZF every n in N is finite.

>

> > Look, presently we work in the system of terminating decimals - by

> > definition. If nothing changes when we switch to the system of non-

> > terminating decimals, do we switch then at all? How could we recognize

> > that we have switched?

>

> I don't have any idea what these questions mean

I know. But it would be nice if you read it again and again. Or try an

experiment: Write a long sequence of digits d_1, d_2, d_3, ... and do

not stop. Are you in danger to leave the domain of finite sequences?

Regards, WM