Re: Well Ordering the Reals

In article <MPG.1dd5363e5b6a4abd98a64c@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <MPG.1dd3e5812d854a2398a62f@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >
> > > Virgil said:
> > > > In article
> > > > <MPG.1dd2ab576b1861dd98a603@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > > >
> > > > > Virgil said:
> > > > > > In article
> > > > > > <MPG.1dd1794bdba1890a98a5dc@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > > > > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > > > > >
> > > > > > > Respond specifically to my comments on Cantor's First.
> > > > > > > Tell me it that, in the context of my well ordering, the
> > > > > > > real number c in the proof does not boil down to the last
> > > > > > > element, after you read my response.
> > > > > >
> > > > > > What sequence does TO alleged 'c' is to be the "last" of?
> > > > > The cmbination of A and B, or either one.
> > > >
> > > > The whole point is that c is NOT in either or in both sequences
> > > > at all!
> > > It is in both, if you allow infinite iterations, as I showed.
> >
> >
> > But the proof, being done in standard mathematics and not in the
> > dreamworld of TOmatics, does not either require nor allow iteration
> > beyond those indexes found in the infinite set of finite naturals.
>
> Then what it really proves is that it requires an infinite number of
> iterations to get to this real number c, not that it doesn't exist.

No one has said that the number does not exist, merely that it does not
belong to the enumeration, so that nothing further is needed at all.

> In other words, some
> real numbers in the set (most, in fact) will require an infinite
> number of bits for their representation.

Which shows, as was to be proved in the first place, that the reals
canot be enumerated by the infinite set of finite naturals.

>
> >
> >
> > > > Since these sequences are mappings from the infinite set of
> > > > finite naturals to the reals, there is no such "after".
> >
> > > Use your standard concept of limit, as above.
> >
> > The standard concept of limit does not have any such "after".
> >
> > Definition: For N being the the infinite set of finite naturals,
> > and R being the set of finite reals,
> > 'lim_{n -> oo} f(n) = L'
> > is defined to mean
> > 'for every positive real epsilon in R there exists an n in N
> > such that whenever m in N and m larger than n, then |L-f(m)| is
> > smaller than epsilon.'
> >
> > Where in that definition does it mention "after"?
>
> It doesn't.

Then why did TO lie about it?
> >
> > > > WE are including the infinitely many finite naturals, which is
> > > > all anyone needs to establish the theorem.
> > > So, it is considered sufficient to enumerate the reals using
> > > COUNTABLY many bits?
> >
> > The whole point, which TO seems to be missing, is that one cannot
> > ennumerate the reals with only the infinite set of finite naturals.
> >
> > One wonders what TO thinks the Cantor theorem is all about if he
> > does not understand even that much of it.

> I am asking exactly what your criteria are for a full specification.

That is not the issue in the Cator theorem. But the least upper bound
property will suffice.

> Obviously, you believe that a countable number of bits is sufficient,

Since a countably number of digits suffice to represent any real, any
real requires only a countable number of bits. That should be simple
enough even for TO.


> I'm going to have to study up.

TO should have done this long since, but better late,...
.

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