Re: An uncountable countable set

*** T. Winter wrote:
In article <ec9r9u$7tb$1@xxxxxxxxxxxxxxxxxxxx> aeo6@xxxxxxxxxxxx writes:
> *** T. Winter wrote:
> > In article <ec8hlf$qq8$1@xxxxxxxxxxxxxxxxxxxx> aeo6@xxxxxxxxxxxx writes:
> > > If the formula applies to an infinite number of finites, then does the > > > sum of the aleph_0 finite naturals equal (aleph_0^2+aleph_0)/2?
> > > > I think there is a misunderstanding. The formula
> > sum{i = 1 .. n} i = n * (n + 1) / 2
> > holds for every finite number n, so it holds for infinitely many finite
> > numers n (as there are infinitely many finite numbers n). But we can
> > not switch to an infinite sum (that is something different).
> > Well, we can. If we turn to what we know about infinite series, we can > apply notions such as, if every term in series A is greater than its > corresponding term in series B, then the sum is obviously greater.

Yes, you can claim that, but you might get in trouble. We may similarly
state that if every term in a sequence A is greater than its corresponding
term in a sequence B, then the limit is obviously greater. Now apply that
to: lim{n -> oo} 1/n > lim{n -> oo} 1/(2n). You have to formalise what
you mean with an infinite sum of a diverging series before you can state
things like that.

If, instead of talking about limits, we apply some unit infinity n, then we can maintain that 1/n>1/2n, one being half the infinitesimal value of the other. We might get in trouble, but I see none ahead, and no one's convinced me there's any down this stream.

So, if we, for instance, say 2n<n^2 for n>2, then oo>2, and 2*oo<oo^2. Transfinitology says that 2*aleph_0=aleph_0^2.


> claim there are infinitely many n e N, aleph_0 of them. So, if that > formula represents the sum of the first x terms, and you plug in aleph_0 > for x to include all of them, then you get that result.

But that is no proof.

No proof according to which set of assumptions? It's something of a proof of contradictions between sets of assumptions.


> > > In > > > standard theory, would this not equal aleph_0, and if so, does it make > > > sense that sum(n=1->aleph_0: 1) = sum(n=1->aleph_0: n), when n>1 for
> > > all n>1? The scond would appear to be clearly a larger sum.
> > > > No that does not make sense. sum{n = 1 .. oo} 1 is not defined, the same
> > holds for sum{n = 1 .. oo} n. If you want to use them you have to
> > provide a definition for them.
> > All that needs doing is declaring a unit oo and allowing it to be used > formulaically. sum{n = 1 .. x} 1=x, so sum{n = 1 .. oo} 1=oo. sum{n = 1 > .. x} n=(x^2-x)/2, so sum{n = 1 .. oo} n=(oo^2-oo)/2. It was perhaps a > year ago that three of us independently said the sum is (|N|^2-|N|)/2. > It's not a problem dealing with infinite values, once you declare an > infinite unit. "Diverges" doesn't specifically describe the value of the > sum. :)

Are you sure there will no contradictions come up? How do I operate with
oo?

I am quite sure. I don't have a full spec at this time of arithmetic operations on Big'un (oo, unit infinity) and Lil'un (unit infinitesimal). Please ask specific questions. That would be helpful.


> > He may be trying to point that out, but not in a way that I do understand.
> > Moreover, I have stated over and over again that aleph_0 does not behave
> > like a normal number. It is just people like WM and you that wish that
> > if behaves like a normal number, but it does not do so. I see no problem
> > with that. You can not find a midpoint in the ordered set of natural
> > numbers using a measure derived from the standard measure of the reals.
> > True. WM and I and others want numbers to behave like numbers, and > cardinalities and ordinals simply do not.

Yes. The only problem you and WM have is with the terminology. But that
is just labels.

You think it's just terminology? :|


> We find them useless and a > digression from the study of quantity and representation which we see as > being the foundations of math.

Yes, you find them useless. Others do not find them useless. Just opinion.

Perhaps. The proof of the tree is the fruit, or the nut. :)


> It irks people like us when set theory > claims to be the foundation of math, and yet makes all sorts of > exceptions and new rules for infinite values.

That is not the case. The rules come from the way these things are
*defined*, not from anything else. Consider Conways surreal numbers,
they act much more in the way of numbers (they even form a field).

Conway seems to have worked hard to make his system not conflict with standard set theory, and yet achieved a lot, in terms of making infinite values somewhat palatable. That's to be commended, but not necessarily emulated.


> It's some kind of logical > construction, but for folks like us, it's really not related to > mathematics.

It is related to mathematics. Perhaps it is not related to physics,
but it is related to computer science. Read about the computable
numbers as defined by Turing.

None of Turing's work concentrated on the transfinite, that I've ever heard. I'd be very interested in even one quote of his on the topic. :)

Tony
.

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