Re: Earth 8??

Nathan Sanders wrote:
In article <LePOf.115941$QW2.61005@dukeread08>,
Ophidian <ophidian23@xxxxxxx> wrote:

Nathan Sanders wrote:

In article <suKOf.115920$QW2.35241@dukeread08>,
Ophidian <ophidian23@xxxxxxx> wrote:

Nathan Sanders wrote:

Give me a real number, I'll give you an integer.
Let's see who runs out first.

The usual proof goes as follows (we'll only consider positive reals between 0 and 1 for clarity; that set is the same size as the set of all reals):

List all of your numbers and my numbers, as follows:

1 0.3857354703483719453...
2 0.9264337854905487822...
3 0.2395549230438505875...
etc.

Now, it's clear that you've exhausted all of your numbers.

No, seriously.
How do I run out?

Because I said "list all of your numbers".

Which is only possible given infinite time.
Or shorthand.
Both of which apply equally to your set.

Which positive integer do you think is missing from the infinite list of positive integers that begins with 1, 2, 3, and infinitely proceeds one-by-one?

That's not a list.
That's a description.

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!

The result is 0.439..., a number that is different from every other number on the list, which means it *can't* be on the list yet, so it must be a new number. You ran out, I didn't.

You just gave 1 more number, and a pattern for generating more.

Not just more... more that are *guaranteed* not to be on a countably infinitely long list.

I just heard a bar move.
Stay with the arguement at hand.

I interate by 1 and keep pace fine.

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

When?

Otherwise, it wasn't a countably infinite list to begin with. That's the whole point; once we've constructed an infinitely long list, all of your numbers are there, but I can still prove that there are numbers not yet on my side. It creates a contraction to assume that I can list all of mine, because I can show that in fact, even with an infinitely long list, there are still lots more numbers I haven't yet listed.

Here's my list:
2
4
6
etc.

That's infinite.
And missing some integers.
But lets make it harder to fill the gaps:
1
4
9
16
etc.

That is what I was trying to demonstrate.
1:1 alone is useless.
A determinitive, predictive rule applied against a known "countably infinite" set is necessary to conclude a new set is also "countably infinite".

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.

"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.

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_.

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.)

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.

Infinity is an abstraction, and it has to be dealt with in abstract terms.

That's why 2 times infinity equals infinity, 2 times infinity divided by infinity equals "undefined". It's the standard boundary problem/singularity issue.

B) A subset contains less elements than its parent set. (For arguement's sake, ignore the special case of a set being its own subset.)

B is false; it's only true for finite sets.

Only if you assume all infinities are of equal size.

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.

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.

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.

(Further, A is only marginally true, because formally, set size is defined without reference to numbers or elements, just equivalence in mappings (one-to-one pairings).)

The size of non-infinite sets is independent of the number of elements contained?

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.

A set has no size when not compared to another set?

Not really, no. Numbers are just special types of sets, so saying that a set has "size 3" is just shorthand for saying that there is a 1:1 mapping between the set and a special set we have denoted with the label "3" (often defined as the set { {}, {{}}, {{},{{}}} }.)

I said that.
You snipped it.

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. One set contains all the elements of the other, plus many elements that the other does not contain. Simple. Both sets are infinitely large. Simple. The conclusion should not be "both are the same size" but that "(X times Infinity = Infinity) tells us _nothing_ about X. Which was part of the point of my 1:1 'matching' exercise. (Does that word work better for you?)

A symbolic paradox is a cute math trick, but it's no more 'valid' than Achilles failing to catch the tortoise.
.

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