Re: Well Ordering the Reals

Virgil said:
> In article <MPG.1dfa3a448e258b0098a7f0@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > Discontinuities are distasteful. Even 1/x is continuous at x=0.
>
> Not outside TO's empty head!
> >
> > >
> > > His difficulty will be how to deal with a that the "carry" operation
> > > that carries out of the neighborhood of one of his "limit" digital
> > > points and into the neighbor hood of another.
>
> > That depends on the connecting bit sequences. It doesn;t break anything.
>
>
> If TO is to have only finitely many digit positions between two of his
> "limit points", he can eliminate either one of them without loss, so one
> can assume that this need not ever not happen.

That's right. You don't define limit points finitely far from any other limit
point.

>
> But if one has infinitely many digit positions between two consecutive
> "limit points" one has precisely the same situation as when one has
> only a first digit and a last digit with no "limit points" between them,
> so that no advantage has been gained.

If the string between them is defined, either with a bit pattern or as equal to
another corresponding string, then no limit points are needed in between. What
yyou seem to be missing is that the left "end" of the string is itself marked
by a limit point. With one point you can define a finite neighborhood. If you
need to define specific finite neighborhoods of points infintiely far from each
other, then you have to establish another local origin, or limit point, to
define the finite neighbothood around.

> > >
> > > For instance, given any one of these "limit" digital points, how does
> > > one determine what the next one will be? There has to be a next one,
> > > doesn't there?
> >
> > You would define a next one wherever the bit pattern changes in one string or
> > the other. If you have N^N 1's, then you just need one extra point at N^N.
>
> But TO cannot even compare bit patterns until AFTER he has chosen his
> "limit points". The only places he CAN compare bit patterns are at
> places only finitely distant from an end digit. For all other
> comparisons, TO must know before comparing which of two numbers is
> larger in order to find where to put his "limit points".

Can you compare two numbers without knowing the digits, or the bit positions?
No? Thought not. Can you describe how to change the timing belt of a car, when
you have no specification as to what kind it is? Would you prescribe medicine
for an ailment, without knowing what it is? Then why do you expect me to be
able to compare incompletely specified infinite numbers, when you cannot even
come up with any specification, and can't follow this scheme long enough to
realize that it works? Haven't you ever tried playing with infinite numbers? Do
you ever play with numbers? Probably just your checkbook, and it has a whole
other column for the cents, or else you'd probably *** that up too.

>
>
>
>
> > >
> > > And there is always the vexing problem of how does the transition from
> > > one such "neighborhood" to the next work?
> > >
> > What do you mean, how does it "work"?
>
> Which word, or combination of words, in "How does it work" confuses you,
> TO?
Which word, or combination of words, in "What do you mean" do you not
understand? Are you incapable of asking a specific question? You probably don't
know what to ask, so you ask some vague nothing, and when I don't answer, you
declare victory. How adept!
>
> > When you compare two T-riffics, you either assume that string is the
> > same in both but unknown,
>
> Given two T-errible numbers having all but one digit zero and that digit
> infinitely far from both ends, without knowing a priori anything about
> the relative positions of those non-zero digits, what is the procedure
> for comparing the numbers for size?

When someone stops you on the street and asks how to get home, but then can't
tell you where they live, or even what their name is, what can you do? Shall I
take you to the local police and let them sort it out for you, and see if
you're missing? Maybe I should just roll you and see if you have any money.

If I say, what is the sum of two finite numbers which each have a 2 in them
somewhere, can you give a coherent answer?
No.

If I say x=2y and y=3z, and then ask what the value of z is, can you tell me?
No.

Then why do you ask stupid questions where you know there is not enough
information to give an answer? Do you think that proves anything besides your
sense of depseration in this discussion?
It doesn't.

When did you stop beating your grandmother, dork?

>
> If no procedure can be found that does not depend on having some a
> priori knowledge of those numbers, then TO's system collapses.
I see. So if I ask you waht the cardinality of set X is, without any definition
of what is or is not in it, you can tell me? Great! What is the cardinality of
set X? If you cannot answer this question, then set theory and all of
mathematics collapses around Virgil's dented head.
>
>
>
> > What exactly are you asking?
>
> I repeat:
> Given two T-errible numbers having all but one digit zero and that digit
> infinitely far from both ends, and without knowing a priori anything
> about the relative positions of those non-zero digits, what is the
> procedure for comparing the numbers for size?
The procedure is to kick you in the nuts and pee on your head while you writhe
in agony from my steel toed purple suede platform knee-high boots. "I'm
singin' in the rain..."
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

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