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Topic: Simple analytical properties of n/d
Replies: 20   Last Post: Mar 11, 2013 11:01 PM

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 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Simple analytical properties of n/d
Posted: Mar 9, 2013 9:28 PM

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}?