|
|
In article <45980b36@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
> Robert Maas, see http://tinyurl.com/uh3t wrote:
> >> From: Tony Orlow <t...@xxxxxxxxxxxxx>
> >> The difference between countable and uncountable is whether
> >> everything is, or isn't, finitely distant from each other thing.
> >
> > That's not correct. Consider for example:
> > -1- Four points where axis cross unit circle.
> > -2- All places on unit circle where x-coordinate is rational.
> > -3- All points on unit circle.
> > In each case, any two points are finite distance from each other.
> > Yet in first case we have a finite number of points, in second case
> > a countable infinity, in third case an uncountable infinity.
> >
>
> In any linear ordering each of those sets, the countable sets will have
> each element finitely before or after each other, whereas in the
> uncountable set, any linear enumeration will result in elements
> infinitely past other elements.
Wrong, as usual. If one "unrolls" the unit circle onto the straight line
x = 1, with the point (1,0) fixed and, for 0 <= t < 2*pi, the point
(cos(t), sin(t)) going to (1,t), then all the uncountably many points of
the circle become ordered linearly with no points infinitely past any
others, and all infinitely many points within a finite distance of each
other..
>
>
> >
>
> It's an attempt at improvement.
And like all of TO's attempts, it failed miserably.
> Set theory was inconsistent for a long time.
Set theory has never been as inconsistent as TO is on his best day.
> >
> > If you combine order and set size, then you are not comparing sets,
> > you are comparing something more complicated than just sets.
>
> Then the naturals are not a set, nor the reals.
As ordered sets they have more necessary structure than mere sets, but
that does to make them non-sets.
>
> > Cantor's cardinality is the best that is possible when comparing
> > just sets. But with more complicated structures there are
> > alternatives that make use of the extra structure. For example,
> > Lebesgue-measurable subsets of the unit interval can be compared
> > per their Lebesgue measure, while non-measurable subsets can't.
> > Subsets of any ordered set can be compared lexicographically,
> > providing that the portion where they disagree has a least element.
> > For example, if we have an ordered-set S such that every subset has
> > a least element, then any two subsets can be compared
> > lexicographically, so we have a total ordering on the power set of
> > S.
> >
>
> Sure, there are quantitative sets and discrete sets of symbols or
> values. A common approach, I think, can handle both better than
> cardinality, when order is taken into account. If we have a notion of
> order on a set, should we ignore it?
If TO wants a theory of ordered sets, he cannot then claim that it
applies to unordered sets.
Cantor cardinality applies to all sets, including unordered ones.
Thus TO's attempted theories are inherently restricted in ways that
Cantor cardinality is not.
>
> >> There's not really an infinite number of finite naturals, ...
> >
> > What's that supposed to mean?? Are you saying there are only a
> > finite number of finite natural numbers? Tell us all exactly how
> > many finite naturals you believe there are.
> >
> > Summary: You're flat out wrong, and I want you come to realize that.
> >
>
> It depends what you mean by "really infinite". There are only a finite
> number of finite naturals between any two others. This is "countably" or
> "potentially" infinite. Personally, I don't consider that "actually"
> infinite. Uncountable is actually infinite.
That TO wishes to impose his own secret meanings for words or phrases
does not mean that anyone else need accede to his peculiarities.
>
> >> The rationals, well, that depends on whether you allow infinite naturals...
> >
> > It doesn't matter whether somebody *allows* something or not. Per
> > the Peano axioms, there *are* more than any finite number of
> > integers, and more than any finite number of rationals.
>
> That really depends on the application of the von Neumann ordinals. If
> you begin the naturals at 1, then the count of n consecutive naturals
> ends at n. If there are aleph_0 naturals, then aleph_0 is a natural.
WRONG! As usual! In order for that to be true one would have to have
aleph_0 be a member of itself, and no set in ZFC or NBG is a member of
itself. So TO's claim requires the impossible.
> So, I reject
> the von Neumann model of the naturals as bunk.
Note that von Neumann, and everyone else, rejects TO's models as worse
than bunk. And with better reasons.
>The set of finite
> naturals is at least as large as every finite, but not larger.
Which finite is it no larger than?
> No
> countably infinite sequence of points can ever achieve any finite
> measure as a line.
In most measure theories the measure of a countable set on a line is
zero, but that does not apply to the span of the set.
>
>
> If infinite naturals are allowed as numerators and denominators, then
> you have an uncountable set of hyperrationals.
Non-existent objects can be given whatever properties one wants.
> >
>
> While it's true that any finite is still finite when
> incremented, this only holds for a finite number of increments. Where
> you increment 0 an infinite number of times, you have produced that
> infinite number. That's not allowed in the set.
Precisely. And doing things that are not allowed is what TO is all about.
|
|