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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. That's what I meant by "finitely
distant" - in terms of intervening elements. In reality, if you want to
talk about sets of points, as you have, any two distinct points have an
infinite number of intermediate points.
If one measure says the two are equal, then that means it hasn't
detected any difference between them. If another measure detects a
difference between them, then a difference can be said to exist. If a
combination of rules says both that A>B and A<B, then there's a problem,
from an order standpoint.
That's correct. Cantor's definition by which sets are compared per
cardinality, is consistent in this way. Everything our latest troll
has come up with is inconsistent in some way.
It's an attempt at improvement. Set theory was inconsistent for a long time.
Is there a way of combining order with set size to create a
measure for finite sets which can be generalized to the infinite
case? Hmmmm.... methinks so.
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.
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?
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.
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.
That makes no sense. The notion of a smallest infinity violates the
basic concept of subtraction as producing a smaller result. So, I reject
the von Neumann model of the naturals as bunk. The set of finite
naturals is at least as large as every finite, but not larger. No
countably infinite sequence of points can ever achieve any finite
measure as a line.
Either of them is a countably-infinite set.
If infinite naturals are allowed as numerators and denominators, then
you have an uncountable set of hyperrationals.
Given a set of axioms, and rules of inference on which the axioms
are based, it isn't a matter of somebody *allowing* a particular
consequence of those axioms. Something either is or is not a
consequence of those axioms regardless of who allows and who
doesn't allow that to be a consequence.
There is also the matter of interpretation of a form of proof, such as
inductive proof. 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.
There is this Twilight Zone between the finite and infinite, no
largest finite and no smallest infinite, ...
That's not correct. There is indeed no largest finite, but there
*is* a smallest infinite, namely Aleph-null. (Well, unless you
allow my concept of Turing-undecideable sub-countable. Do you?)
I don't know about that, but if x-1<x is to have any consistent meaning,
aleph_0 is bunk. It's a phantom, like the largest or smallest finites,
or the largest infinitesimal. It doesn't exist.
all infinite larger than all finite.
Correct.
Omega is a phantom, ...
Wrong. Omega is an ordinal, an equivalence class of ordered sets
modulo mappings that preserve the order. Aleph-null is a cardinal,
an equivalence class of sets modulo *all* 1-1 mappings. Several
different ordinals, such as Omega+1 and Omega+Omega, all have the
same cardinality. They are different ordinals because there's no
order-preserving mapping between them, but they have the same
cardinality because if you ignore order there *is* a 1-1 mapping
between them. For example:
Omega + 2
a1 a2 a3 a4 a5 a6 ... b1 b2
b1 b1 a1 a2 a3 a4 a5 a6 ...
Omega
Notice how I moved b1,b2 from after all the a's to before the a's?
But there's no order-preserving mapping between Omega+2 and Omega.
Try to find one if you don't believe me.
That's all very well and good. I understand the standard position on
that. I just disagree that it has any real value, and recommend a system
wherein order is taken into account where it exists and where proper
subsets are smaller sets. Order is imposed on standard sets through the
naturals, with the addition of the primitive order operator,
successor(), or '<'. Set membership alone does not allow us to compare
most infinite sets with any precision. Where we have order, we should
use it. Call it "sequence size" if you want, but then don't call the
naturals a set. You can't have it both ways, can you?
Tony
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