
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}?

