Re: Well Ordering the Reals


Tony Orlow wrote:
> Virgil said:
> > > Countably many bits means
> > > uncountably many strings.
> >
> > And your point is?
> That none of my bits are in infinite positions, and yet the set covers all the
> reals, in my opinion, at least until proven otherwise.

You're trying to construct a well-ordering, right?

You seem to be confused between bijection and well-ordering.

It's certainly true and easy to prove that the countably-long bit
strings,
in which every bit is in a finite position, have the same cardinality
as the reals. It's easy to construct a bijection.

But I gather you're trying to use that to claim you have a
well-ordering
of the reals, and that you don't have since you have infinite
descending sequences.

We know that there exists a well-ordering of the reals. But we also
know
that the order won't be the usual "<" relationship. It will be based on
some relationship which says in many cases that x comes before y
even though x > y with the usual meaning of ">" .

- Randy

.

Relevant Pages

  • Re: Well Ordering the Reals
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