Re: Earth 8??

Jay Rudin wrote:

"Ophidian" wrote:


Prove that your number is missing (I was able to prove it with mine).

Hardly.
You didn't list _all_ your numbers.
If you did then your new choice wasn't missing.
This is simple, c'mon!

No; he's exactly right. He's giving the standard proof that the real numbers cannot be put into a 1:1 correspondence with the integers. All mathematicians accept this proof. The problems you're bringing up come from the fact that you're using common English words that have common English meanings that don't apply to transfinite numbers.

Intentionally.
Which is fair game since no one has offerred up the mathematical definition of "correspondance".
The standard English definition in insufficient to the 'prove' the proofs given.

Speaking. He's right again. Two infinite sets have the same cardinality if and only if they can be placed into a 1:1 correspondence.

Again, new word added, and non-standard definition in use.
Substituting 'cardinality' places us in a more formal arena than 'size' did.

I can't give the proof in rigorous fashion through this forum, but it's proven and accepted by all mathematicians who've seen it.

As and admission of intellectual sleight of hand (call it dishonesty if you choose), I'd agree that the proof (and I have seen it) does demonstrate what _you_ say it does, _if_ you are willing to narrow the defenition of "correspondence" to something resembling my "predictive, determinitive" phrasing. Nathan was unable to do that. You're farther towards demonstrating what the proof _actually_ demonstrates in one post than he was in many. Simply because you are choosing to state most of your definitions and avoid my 'common English' terms.

If you have a countably infinite set and an uncountably infinite set, they cannot be put into a one-to-one correspondence. *By definition* of "countable".

Name one from mine, then one from yours.
Repeat.
Every element of mine will eventually corresponds to one element of yours, with NONE MISSING!!!
Simple.
By definition of _infinity_.

Unfortunately, this is not true, by any mathematical definition of infinite numbers.

Then what you are claiming as a mathematical definition does not match the common English definition. In other words, you guys are going for the "for certain values of 'infinity'..." exception.
I'm trying to stay with the "if the model doesn't fit the observation..." point of view.

You're conflating countably and uncountably infinite sets together.

Duh!
To show that the difference isn't just correspondence and isn't just mapping. (Using non-technical definitions of correspondence and mapping.)

The difference is *precisely* correspondence and mapping, using rigorous mathematical definitions.

Thank you.
Nathan was unwilling to recognize that, though I repeatedly pointed it out.

As I said, pick one from mine, and one from yours.
There's a pair.
Continue until one set runs out.
It doesn't happen, ergo every element of one set CAN be paired to an element of the other set.

Don't confound mathematical abstraction with physical reality. Of course we would never run out in the real world, because human beings don't live forever.

Even if we did, we'd never run out.
By definition of infinity.

I don't need forever to place hem into a 1:1 corrspondence. "I map every positive integer onto its square." I have now mapped all of the innfiite set {1, 2, 3, ...} onto its proper subset {1, 4, 9, ...} I've run out. There are no more positive integers left unmapped. I've mapped 111,111,111 onto 12,345,678,987,654,321, even though I didn't count that far. Every single one is done.

Exactly. Well done.

No, the problem is my not knowing your definitions when you don't even include them! Addition problem of your model not matching the observable.

(Or perhaps creating your own definitions,
which you're free to do of course, but you need to realize that your definitions are non-standard in mathematics.)

And mathematics definitions are non-standard in other scientific fields, lay speech, philosophy, and pretty much anywhere but pure mathematics.

Yup. I won't use mathematics to fix a pipe, and I won't using a wrench to discuss a mathematical issue. Mathematical definitions are the ones to use when discussing infinite sets.

We were originally discussing how 'infinite' world would function in the DCU. A _very_ non-mathematical subject.
(Hey, I've used math to fix pipes! <g>)

In English, we often call that counting.
It disappears in the abstraction by an illusion.
Match element one of set A with element one of set B.
Match element two of set A with element two of set B.
etc.
Where "element one" is the first element chosen from a set, "element two" is the second chosen, and so one.

Actually, the mathematical distimnction is clear, but hard to explain in English.

Understood, but I'm going to take a tangent here for a second.
Einstein made a statement along the lines of "if you can't explain a concept to a lay man, then you really don't understand the concept".
I've seen this borne out repeatedly. I've had college professors who could explain the most arcane concepts; I've had others who could explain basic circuit theory. I've grasped a large ammount of multidimensional geometry, and of quantum physics, all without specific background in those fields. This was done by listening to people, who at least intuitively understood what Einstein was claiming. Hell, I'm currently teaching a 10 year old algebra despite claims that she doesn't have the base knowledge for it. And while she's very bright, she certaintly isn't Einstein. I'm willing to believe you are able to explain things in such a fashion, so long as you remain specific with your terms, and don't claim the 'lay' terms are irrelevant.

But what you start to do with this last quote, is what Nathan eventually did in frustration. It's called special pleading. In this case "I know more about this than you, so just believe me." It's simply not a valid proof of anything. Expert thought in _every_ field has been wrong at one time or another. Now what you are trying to convince me of IS true, assuming you limit your terms. Nathan failed to do so and created arguements that are directly at odds with observation.

You guys also don't get to pull "For values of 'green' where 'green' equals 'blue', the sky is 'green'."

Counting is a process that you do one number at a time. You can never finish doing it.

Exactly.
But you have set the pattern and can project it, just as you can project the elements of 1, 2, 3,...n-2, n-1, n as n approaches infinity.

A 1:1 correspondence can be defined for all numbers at the same time. I define, for example that each positive integer n is mapped onto the product 2n. This defined mapping puts the positive integers and the even positive integers into a 1:1 correspondence. I didn't count them -- that takes forever. I defined a mapping, and I have finished.

Using your non-standard definitions, granted.

Really this whole drift boils down to "mathematical abstraction can show that unequal things are equal". Telling us the set of all integers is the same size as the set of all squares of integers does not make it true.

No. The mathematical proof of establishing a 1:1 correspondence makes it true.

Same 'size'?
Careful.

You are making the unwarranted assumption that the standard properties of finite numbers apply to transfinite numbers.

Actually if you paid atention I made two conflicting arguements.
One depends on the properties being the same.
One depends on them being different.
Both are true as stated.
But let's flip your statement.
You are making the assumption that an established pattern will 'break' when subjected to infinity.
But why do that?
Because a trick does show a break?
Wouldn't that just as easily be believed to be the trick 'breaking'?

Let's go with an example here.
Set A is defined as all positive integers up to n.
Set B is defined as all positve integers up to n and their negatives.
If I populate the sets discretely, I get:
n Set A Set B
1 1 1 -1
2 1 2 1 -1 2 -2
3 1 2 3 1 -1 2 -2 3 -3

Every step of the way Set B will contain twice the elements of set A.
Set B's 'size' is 2n elements.
Set A's is n elements.
The 'size' of set B relative to set A is 2n/n, or 2.
Set B has twice the elements of set A for all discrete values of n.

Now let's choose the highest possible n.
That's layman's infinity.
Now what is the 'size' of set B relative to set A?
Nathan continuously insisted that is equal.

Because of a cute trick you can pull with the math.
All that he could really demonstrate is that "all infinities are of 'equal size'", which is my 'imaginable animal' scenario and my "I pick one, you pick one" scenario. That his pick is based on a calculation simply proves this faster for the limitted subset of "countable infinites".

Now my example above is a simple demonstration that two "countable infinities" do not have the same 'size'.

Hm, in standard English I've just argued that 'all infinities are of equal size' and that 'some infinities differ in size'.

Can both really be true?
Well, actually, in standard English, yes.
That's the root of some great 'paradoxes'.

It's also why in _applied_ mathematics we generally define infinity as 'undefinable'.

But to wrap it up, thanks again for the better defined view of the special case.
.

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