Date: Mar 9, 2013 9:28 PM
Author: William Elliot
Subject: Re: Simple analytical properties of n/d
On Sat, 9 Mar 2013, Ross A. Finlayson wrote:

> On Mar 8, 9:31 pm, William Elliot <ma...@panix.com> wrote:

> > What's R_[0,1]? So now your considering any function f:N -> [0,1]

> > and want to discuss rang f.

> >

> > > Obviously and directly as N is countable, ran(f)

> > > is countable,

> > > Is ran(f) = [0,1]?

> >

> > Of course not, you showed that range f is countable and

> > unable to be any uncountable set.

>

> Apply the antidiagonal argument to ran(f).

Why?

> the only item different from each is and, ran(f) includes 1.0.

Huh?

> Apply the nested intervals argument to ran(f).

What's that?

> The interval is [.0, .1], there's no missing element from ran(f)'s

> [0,1].

>

> The antidiagonal argument and nested intervals argument don't support

> that ran(f) =/= [0,1].

>

> In fact, remarkable among functions N -> R, is that the antidiagonal

> argument and nested intervals argument, DON'T apply to f.

It doesn't? What if f:N -> [0,1] is the constant function f(N) = {0}?