Re: abundance of irrationals!)

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?

> 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.

> 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.

.

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