Re: Well Ordering the Reals

Tony Orlow wrote:
David R Tribble said:
David R Tribble said: 
But this is still much too short to be an uncountably infinite list,
and therefore it cannot possibly denumerate all the reals.
Tony Orlow wrote: 
Actually untrue. Consider this. We have a countably infinite number of
iterations starting with 0. We generate 2^n elements at iteration n. Therefore,
with aleph_0 such iterations, we generate 2^aleph_0-1 elements, an uncountably
infinite set. Is there a rule that says we must generate only one element at a
time? My set produces 2, so it grows like the power set.
David R Tribble said: 
It really is annoying that you don't have a clue about infinite series
or the actual meaning of Aleph_0.  You are confusing arithmetic
exponentiation and the cardinality of powersets - they are not the
same thing, you know.
Tony Orlow wrote: 
It's annoying that you can't just address the issue without an insult. It's not
my fault that your system doesn't work right. Exponentiation is exponentiation.

Heh.  No, ordinal exponentiation is not the same as cardinal
exponentiation.

It's provably true that Aleph_0^2 = Aleph_0, for instance.
That's patent nonsense. If it's provable in your system, the system sucks.

In your opinion. That's fine. But make it clear that your system is entirely different from the standard mathematical one. This might be made easier by introducing new terminology; or at least by making it clear that old terminology was not intended to 

If set theorists didn't want the power set confused with exponentiation, they
should have made upa new symbols, but I guess they had too many already.

Point taken.  The fact that cardinal exponentiation is close to ordinal
exponentiation is the probably the reason that mathematicians use
the same notation for both.  But the fact that cardinal exponentiation
is not identical to ordinal exponentiation can be a big source of
confusion, as you illustrate.  So yes, it might have been better if
they had chosen a different notation.
Actually, the best approach would be to have exponentiation, and if you have some other unrelated function, call it something else. 

That's precisely what's being requested of you, as I understand it.

It's kind of like the
claim that cardinality=size, but then again, not really the same. One really needs to try to keep their terms to a minimum possible size for the problem at hand, not randomly generate lots of extra terms and special rules and exceptions. The power set is 2^n for set of size n, and that means actual, real, normal exponentiation. There is no need for special eponents for cardinals, and any apprent need for it is the result of a kludged system. 

The fact is, the power set, like the binary strings, DO comprise a set witha
relationship of 2^n elements for every n elements in the root set or n digits
in the binary string. For you, 2^aleph_0 doesn't really mean 2^aleph_0. For me,
it does. There is no reason to make the distinction.

Except that the math does not work right if you don't make that
distinction.  For us, 2^Aleph_0 is just a bigger cardinal than Aleph_0.
Then call it aleph_1, if you don't actually mean 2 to the aleph_0 power. What is that exponentiation doing there anyway, if you don't think exponentiation has anythign to do, really, with the relationship? The fact is, it does, and it should be there, and it should be interpreted as exponentiation in the usual sense. That works just fine, as long as you're not shying away from actual infinities. 

They do "actually mean 2 to the aleph_0 power".

But it's provably true that 3^Aleph_0 = 2^Aleph_0, for instance, so
again it's obvious that cardinal exponentiation is not the same as
ordinal exponentiation.
That should not be true, although I think it is in standard theory. 3^n>2^n for n>0, and aleph_0 is certainly bigger than 0, is it not? These are precisely the types of expressions I want to see distinguished, not all lumped together as if arithmetic becomes meaningless for infinite values. It doesn't. 

No, it doesn't. But it does become different. I don't see anything wrong with that.



The infinite sum
  s = 1 + 2 + 4 + 8 + ...
is a sum composed of a countably infinite number of finite terms,
thus it can be said to be a countable (denumerable) infinite number,
of the same order as Aleph_0 (or omega).  Since each term
represents the number of points your mapping defines at each step,
we conclude that your mapping defines only a countably infinite
number of points in total.

And yet, if there are aleph_0 terms, this sum is one less than 2^aleph_0, and
this being the count of elements in the set, it would appear to have a
bijection with P(N) and to be an uncountable set.

Your cardinal math is incorrect.  The sum is Aleph_0 (omega), even
though there are Aleph_0 terms.  Infinite numbers don't behave like
finite numbers.

Yeah, so I've heard, over and over and over, here. It's nonsense to justify some rule with "why not?" "Infinite things need not behave like finite things" is no excuse for making up whatever foolish rules you want, just because no one will ever actually apply them to anything. If you have two infinite series, and one has tems that are always larger than the corresponding terms in the other, then the sum of the first is larger than the sum of the second. If you have sum
(n=1->oo: 1), it is smaller than sum(n=1->oo: n). The first is oo. The second is 2^oo-1. This is PROVABLE inductively.

*What* is provable inductively? According to my calculus book, the statement you offer about two infinite series apply only to series both of which converge. A series which does not converge, regardless of the value of its partial sums, is simply said to diverge. It's unfortunately the case that our notation for this uses the symbol for infinity; but infinity doesn't really enter into it.

> Any system which denies the inequality
there is bonkers, which is the case with transfinite set theory.

Again, you appear to fail to understand that inductive proofs apply only to finite sets/numbers.


So, in your mind there is a difference in cardinality between the power set of set S with size N, and the set of binary strings of length N? Funny how a difference of 1 can change an infinite cardinality, eh? You know, funny in a ha ha sense. More of a HAHAHA AAAHAHAHA HA HA HA HA WAHAHAHAAAAA Oh my god!!! kind of sense.

It is provably true that Aleph_0 - 1 = Aleph_0, for instance.

You can use that word, "provably", until it's the only word on the page. I don't care. I have made myself clear, have I not, that a system that "proves" such things is useless in my estimation?

Again, that's fine. You can hold anything you want in whatever esteem you like. That doesn't prevent others from being able to use it.

> It has no applications, can not be
tested, and fails to jibe with any other approach to these questions. I can "prove" that 1-1=1, too. But, as you are well aware, that proof involves a hidden division by zero. Your system includes such hidden problems that give rise to such faulty results.

To date, you haven't (as far as I've been aware) been able to point out anything that would be commonly considered a problem. That you consider it a problem is irrelevant.



Your mapping function does not "generate" anything more than
a countably infinite number of points, which is far too small to
cover the reals.

Okay, then 2^aleph_0 is the size of a countable set, and Ross is correct in
saying that all infinite sets are equivalent and countable. If a countably
infinite set of countably infinite sets contains a countably infinite number of
elements in those sets, then the only thing uncountable is the continuum, not
the power sets of countaboly infinite sets.

You're confusing things.  Your mapping defines only Aleph_0 points,
no matter how wide the bifurcations are at each step.  Every point
you define can be denumerated by a natural number, so there are
at most Aleph_0 points defined.
Every level is denumerated by a natural number, but there are 2^n elements at level n. At level aleph_0 there are 2^aleph_0 elements. 
The continuum, c, is uncountable, and it is provably true that card(c)
and card(P(N)) are identical.  It is also provably true that P(N) is a
larger set than N.
Actually, it is not true that the power set of the naturals describes the continuum. I know in standard theory it is understood that way, but other approaches show that this is simply not true.

Not quite. Other approaches *may* show (I'm not familiar with any others) that the statement is not true *given that approach*. They don't show that the statement is not true.

> For any value range from 1 to x,
the number of elements of the form log2(n) for n in N is 2^(x-1), or 2^x/2. Over the entire value range of the real line, therefore, there are 2^aleph_0/2 log2's. Now picture the logs base 2 on the real line. Does that look like it includes every real number? Half of them? Not by a long shot. If N is the number of naturals, 2^N is not the number of reals in any real sense, at least outside of transfinite set theory. 
You see why we keep saying that cardinal exponentiation is not
the same as ordinal exponentiation?

I see that it's the case, but I don't see any justification for it. It's wrong, as far as I can tell. 

What, precisely, makes something wrong?

Matt 


You really ought to learn the terminology, lest some will think you
are talking out of your hat.  Start with "countably infinite".
Countably infinite means there is no end to the set, but all elements are in
finite positions with respect to the initial element.
I'll be generous and give you half a point for that.  It means that
every element can be denumerated by an ordinal, i.e., a counting
number.
You mean a successor ordinal, not a limit ordinal, right? 
Each denumerated element involves the successor ordinal of the
ordinal denumerating the previous element.  But the total count of
elements is a limit ordinal.  (But I don't claim to be an expert.)


Isn't c related to an ordinal? 
Since c is a cardinal, there must be a corrresponding ordinal for it.
However, there can be an infinitude of ordinals between two
cardinals.
And we don't know what ordinal it corresponds to, how many are above or beneath it, whether there is anything between that cardinality and that of the naturals, or whether it is aleph_1. It's really hard for me to even fathom how anyone can be satisfied with this system, and why no one has suplanted it with something better yet. (sigh) Of course, people still march to war, gobble down prepackaged garbage, argue about sports, ignore the biggest crimes, and fight about broccoli florets. Such is life. 


.

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