Re: Well Ordering the Reals

In article <MPG.1dd2b585c5bbf1b998a60a@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <MPG.1dd1a3f75d2dc54298a5f5@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >
> > > Virgil said:
> > > > In article <MPG.1dd1575c3f870f3f98a5c5@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > > >
> > > > > Virgil said:
> > > >
> > > > > > In TO's infinite binary strings, I presume that those strings
> > > > > > with only finitely many 1's are ordered like the binary
> > > > > > integers, so that leaving out zeros to the left of all non-zero
> > > > > > digits we can assume 1 < 10 < 11 < 100 < 101 < 110 < ... in his
> > > > > > system.
> > > > > >
> > > > > > Then one can ask which is larger ....101010 or ...010101 ? And
> > > > > > by what rule does one decide?
> > > >
> > > > > Well, for one thing, the first is negative and the second
> > > > > positive
> > > >
> > > > In the ordering 1 < 10 < 11 < 100 < 101 < 110 < ..., which I
> > > > indicated, how does anything become negative?
> >
> >
> > > You didn't read my web page.
> >
> > Why should I read that garbage.
> >
> > I am talking about an earlier ordering of TO's alleged *N, in which the
> > strings having a "most significant digit finitely many placess from the
> > right end are all ordered like binary representations of the naturals.

> So, we are talking about a well ordering, and you are talking about the
> bijection with the power set. No wonder you make little sense.

I am talking about whether TO's "Set" of "infinite strings with both a
beginning and and ending" even exist, much less can be well ordered.

For any well defined string, there must be a serially ordered index set
to designate character positions in that string.

What is you index set, TO? Does such a thing even exist?
And if, as I suspect, it does not exist, that makes all of TO's
creations mere daydreams with no mathematical significance.

> > You, TO, have explicitly called sets finite that Dedekind calls
> > infinite, including the infinite set of finite naturals, which is as
> > countably infinite as they come.

> Right
> > > >
> > > > Both of Cantor's proofs that the cardinality of the reals is a
> > > > 'larger' cardinality than that of the infinite set of finite
> > > > naturals are quite valid using the standard (Dedekind) definitions
> > > > of finiteness and infiniteness of sets.
> >
> >
> > > I don't argue with that result, but I do with many others, and I do
> > > with the method.
> >
> > TO is in o position to chop logic until he gets cured of his QD problem.
> >
> >
> >
> > > > Thus no merely Dedekind infinite enumeration of the real is
> > > > possible.
> > > You mean a countable set? I already agreed to that. It requires an
> > > infinite number of iterations to enumerate the infinite set. That's
> > > what makes it infinite in my book.
> >
> > Then it requires an infinite number of iterations to iterate the
> > infinite set of finite naturals. That's what makes the infinite set of
> > finite naturals infinite in everbody's book.
> >
> > > Countable sets are limited to finite iterations.
> >
> > But not finitely many iterations.

> Yes, finitely many, as the number of iterations to reach any finite
> natural is finite.

TO is off in his dreamland again where endless sequences all end.

> >
> > So of what "set" does TO allege c is a last element?
> The set of reals, given my well ordering.

But TO's alleged well ordering of his alleged set has been shown to
allow endless descending sequences, i.e., non-empty sets with no
smallest element, thus violating the very definition of well- ordering.

TO loses again!
.

Relevant Pages

  • Re: infinity
    ... >> Tony Orlow (aeo6) wrote: ... >>> infinite value, then I am only taking a finite number of steps. ... > completed an infinite number of iterations, ... >> finite naturals, where each step takes a constant finite unit ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... if I'm blind? ... any subset that contains an infinite natural is not ... >> that contain only finite naturals. ... But cannot be uncountably long and be bit STRINGS, ...
    (sci.math)
  • Re: infinity
    ... If the set of all finite naturals is a finite set then that sum MUST ... > be infinite unless that length is allowed to be infinite. ... No one disputes that any set of strings of bounded length and bounded ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > infinite, including the infinite set of finite naturals, which is as ... >> You mean a countable set? ... >> infinite number of iterations to enumerate the infinite set. ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... > Okay, so you cannot find the largest element of N, because you cannot ... > iterations to complete the process. ... I claim that a counting process over the elements of ... that it has only finite size unless I discard infinite ...
    (sci.math)
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