Re: abundance of irrationals!)

Russell said:
> aeo6 Tony Orlow wrote:
>
> [snip]
>
> > Excuse me, but the nodes of an infinitely deep binary tree can
> obviously be
> > enumerated in a linear manner, corresponding to binary integers and
> thus to the
> > naturals.
>
> Yes.
>
> If you want to complain that it would require infinite digits for
> > most values,
>
> Huh? What does that even mean? What values are you
> talking about? And digits of what?
Duh. The digits of the binary number that represent each node on the tree,
where each bit represents a right or left branch as a 0 or 1. Infinite strings
of such digits represent elements infinitely far out on the branches. What did
you think, when I referred to correspondence between the nodes of a binary tree
and binary integers?
>
> > that's irrelevant. it's still an enumeration of the reals.
>
> No. The obvious linear enumeration of *nodes* is not an
> enumeration of the reals. For that, you need to enumerate
> the different paths. That's not merely a rearrangement of
> the enumeration of nodes; if you think it is, then please
> tell us how you associate each node with one and only one
> path. Can't be done.
Each bit enumerates a step in the choice of path. Think about it.
>
> > Enumeration is irrelevant to the level of infinity, except insofar as
> we must
> > rearrange the quantities in order to enumerate them linearly. Don't
> look in
> > your bag of axioms for proof of this. It's been conveniently
> excluded.
>
> Yes, it would be rather inconvenient to make that an axiom,
> since it's inconsistent with the other axioms. I think you've
> gotten stuck thinking about rearrangements that turn out not
> to exist in fact.
>
>
Think what you like.

--
Smiles,

Tony
.

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