Re: Galileo's Paradox and the Project of the Reals

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.

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.

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.
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.

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.

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.
Either of them is a countably-infinite set.
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 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?)

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.
.

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