Re: Earth 8??


"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.

I interate by 1 and keep pace fine.

No, because you've used up all the integers.

When?

When he defined the correspondence. He has demonstarted that for *any* 1:1
correspondence, there will be numbers left over.

A 1:1 mapping *IS* a "determinitive, predictive rule"! That's
essentially the definition.

You added a word again.
I contradicted the original definition AND asked for a correct one.
You then act as though the correct one has been used all along.
Of course you won't follow my points if you can't even communicate yours!
I'll let you go and take it up with someone who understands the math, the
language, and the realities.

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. The rational
numbers, the integers, the positive integers, the even integers, the
positive integers divisible by 100 are all of the same cardinality, even
though each one above is a proper subset of the one before it. These sets
all have cardinality Aleph-null. The real numbers have cardinality
aleph-one, and it cannot be put into such a 1:1 correspondence.

"That's what a 1:1 correspondence means: that _every_ member of one set
can be paired up with a member of the other set with none left over."

That CAN be done with any two infinite sets.

No, it cannot.

Reread it.
Carefully.
I need not repeat it again.

No, you needn't because you are mistaken. Any countably infinite set
(aleph-null) can be put into a 1:1 correspondence with any other countably
infinite set. Any set of cardinality Aleph-1 can be put into a 1:1
correspondence with any other set of cardinality aleph-1, but not with any
countably infinite set, nor with any higher-order transfinite number
(aleph-two, aleph-3, etc.)

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.

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.

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.

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.

No. In fact, that's precisely the opposite of what I've said multiple
times. Countably infinite sets are all the same size.

I have shown that they aren't.
Your abstraction does not meet the reality.

Yes, his abstraction is exactly correct, by the definitions of cardinality
for inifinte sets.

Two mathematically infinite sets have no requirement to be of equal
actual size, even if they are both "countably infinite".

If they are countably infinite, yes they do, by the very definition of
countably infinite.

I'm seeing a pattern here: your problem is not knowing the definitions of
the terms you're using.

You are a mirror onto reality. All the details correct, and all of it
exactly backwards. He knows, and you do not, the definitions of the terms
he's using.

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.

No... the notion of "set" and "size" is independent of the notion of
"number". Indeed, mathematicians take sets to be primitives, with
numbers derived from them. Thus, the sets {a,b,c} and {d,e,f} can be
shown to be the same size without doing any sort of counting or use of
numbers. We say they are the same size because there is a 1:1 mapping
between them: {(a,d), (b,e), (c,f)}.

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

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.

If you really are interested in learning about infinity (rather than, in
typical Usenet fashion, just arguing for the sake of arguing and stubborn
unwillingness to admit that you don't know the material you're talking
about),

Ding, I was hoping I didn't go there first above. ;)

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. You are making the unwarranted assumption that the standard
properties of finite numbers apply to transfinite numbers. This is simply
untrue. Just as we lose the property of "well-ordering" when we expand from
the real numbers to complex numbers, and we lose the property that every
subset has a lowest number when we move from the positive integers to the
integers, we lose some properties when we move from finite to transfinite
numbers.

The properties you are assuming to be always true are only true in the case
of finite numbers.

John (Jay) Rudin, Ph.D. (Operations Research), University of Texas at
Dallas, 2000


.

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