Cantor and Numbers Larger than Infinity

[TechieGuy]

Oh man think about this let us agree we have a number line that counts 1,2,3,4… ∞

Let us agree to start at ∞ and count backwards. By the time we reached 0 we would have counted a finite ∞

That’s not how ∞ works…

 

[catholicray]

Would you mind explaining yourself here.

 

[TechieGuy]

Infinity is not a number. If it were a number, than infinity plus one would be a larger number than infinity.

 

[catholicray]

I agree with you but I don’t see why this poses a problem for the understanding. The numbers between 0 and ∞ are infinite.

Here let me explain it another way. Suppose you start at (-∞) and start counting toward 0. Basically by the time you get to zero you would have counted an infinite amount of numbers.

For the sake of thought let us agree we can not count the numbers are we still unable to conclude that the numbers from (-∞) to 0 are infinite?

 

[tafan2]

You can’t start at - infinity, there is no such ooint on a number line.

 

[Dovekin]

Suppose you start at (-∞) and start counting toward 0. Basically by the time you get to zero you would have counted an infinite amount of numbers.

If you start counting at (-∞) you will always get to (-∞). Counting does not get you closer to 0.

 

[TechieGuy]

I’m pretty sure you understand what you mean to say, but I’m not getting what you are trying to convey

 

[catholicray]

You can’t start at - infinity, there is no such ooint on a number line.

I think we are capable of observing precisely this point on a number line. It depends on our approach.

If you start counting at (-∞) you will always get to (-∞). Counting does not get you closer to 0.

Ah your precisely right and yet we count finite infinities all the time. Let’s see if I can explain myself better…

The simple number line:
1,2,3,4,5……(∞)

Now you may disagree with the above model but on our simple number line you and I agree that if we wanted to we could count forever I could fill every moment of the rest of my life with counting because there are ∞ numbers on that number line. Now bear with me because I’m trying to get my hands around this monster.

Let’s examine a couple more number lines to paint a picture.
[0] 1/2, [1]
[0] 1/4, 2/4, 3/4, [1]
[0] 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, [1]

I’d like to look at this one more way. Let’s get really small and then really large really quickly.
0
0.0000000000000000000000000001001010010………
0.0000000000000000000000000001101001010………
0.0000000000000000000000000010100100100………
and also:
0.9999999999999999999999999999999999999………
and finally
1

Two observations:
[1]Counting to 1 is counting a finite infinity. Huh?
We’re just going to move on for now.
[2]If we agree that a number like 0.0000000000001001 has value then the available value between 0 and 1 is an infinite infinities.

In a very real way we are counting by ∞ every time we count but we ignore this fact.
Earlier I suggested that if we count backward from ∞ to 0 then 0 would represent ∞ and I stand by that but to account for that we have to view our number line differently where the count is really just
∞,∞,∞,∞,∞,∞,…….

0 and 1 and so forth are basically placeholders for ∞

Here’s another view of the number line to consider:
1∞,2∞,3∞,4∞,5∞……(∞∞)

In this view we can see that ∞∞ represents an infinite infinities. This is however redundant and only useful as a visual aid because technically ∞ is a simplification. A number that already represents an infinite infinites.

All of this to say that we can view the count from 0 to 1 on a number line as an infinitesimally small count or the count from 0 to 1 is one of infinite infinites.

And so there are infinitely larger infinities - Cantor
and the largest infinity is present in the space between 0 and 1

Therefore, I said Cantor was absolutely right and infinitely wrong.

 

[tafan2]

You are close, but your conclusion is off. Yes, as a professor said once, we can do an infinite number of steps in a finite time. If I walk half way to a wall, and the walk half way again, and repeat, I never quite get there? Wrong, if I walk to the wall, I walked half way, and infinite number of times.
What does that tell us? There are two sizes of infinite sets: countable nfinite and uncountable infinite (ie the continuum). One is the size of the set of natural numbers, the other is the size of the set of real numbers.
There is an uncountable infinite number of real numbers between any two real numbers. So we can say that the number if real numbers between 0 and 1 is the same as the number of all real numbers.
Likewise, the number of natural numbers is the same as the number of integers and is the same as the number of rational numbers.

That’s all, it doesn’t say Cantor is right and wrong, it says he is right.

 

[catholicray]

I disagree with you. You suggested that you can not count to ∞ and what I am suggesting is that we do it all the time. 0 to 1 is a finite ∞

The reason you can walk to the wall is because you have traversed a finite ∞

An utter paradox!!!

Even so this is why I say that Cantor is wrong and right. There are infinitely larger infinities and the largest infinity is contained in the count from 0 to 1

 

[tafan2]

No, I said the same as you, we do infinite number if things all the time, that’s why I gave my example. But this is NOT a paradox. It simply illustrates that there are two types of infinity, when you look at the size of a given infinite set. Countable and the continuum, the number of natural numbers and the number of real numbers. There is nothing paradoxical about saying there are more real numbers than natural numbers. It’s actually quite obvious.

It’s actually countable sets that seem less intuitive. The fact that the number of rational numbers is the same as the number of natural numbers.

 

[catholicray]

I definitely get what your saying but that conclusion is based on our agreeing that the value between 0 and 1 is 1 rather than a place holder for ∞ (infinite infinites)

The reason we are concluding that there are more real numbers than natural numbers is because you assign the value of 1 to the natural number 1 and then examine the number line at it’s full potential. Consequently you like Cantor are right and wrong…… not or wrong but you’ve managed to make a statement that carries both values at the same time. A paradox.

Reviewing quickly
I stated a ago that the value of 1 should be relative to the subset of numbers you are viewing. Therefore when Cantor looks at a lower subset 1/2, 1/4, 1/8, etc. the value of 1 increases relative to that subset.
1/2 and 1 = 2
1/4 and 1 = 4
1/8 and 1 = 8
and
1/∞ and 1 = ∞

or rather (in our imaginations now) you wouldn’t go from (∞-1) and then count the next number as 1 lol accept we do this all the time rofl

Again you can imagine how Cantor is wrong by using multiple dimension where all of our natural numbers to ∞ fit between the space of 0 and 1 on a higher dimension. Wait there is more. We have to say that the natural numbers on a higher dimension are also numbers that exist.

I don’t know if I’ll ever convince you or anyone for that matter and that’s fine but paradoxically Cantor is wrong and right at the same time.

That is I agree with you that absolutely you are right there are more real numbers than natural numbers (bing bing bing bing bing we have a winner!!!) and by the way I absolutely disagree with you.

 

[catholicray]

By the way all this talk of infinity is making me tired lol

 

[tafan2]

I definitely get what your saying but that conclusion is based on our agreeing that the value between 0 and 1 is 1 rather than a place holder for ∞ (infinite infinites)

Ok, its actually rather impressive that you are figuring this out on your own (although you are not quite there), but once someone does this, they should start using standard terminology. It makes communicating very simple. Your term “infinite infinities” is really just a way of saying the more standard term “uncountably infinite”. And this term is actually more descriptive, as I will try to illustrate.

The reason we are concluding that there are more real numbers than natural numbers is because you assign the value of 1 to the natural number 1 and then examine the number line at it’s full potential. Consequently you like Cantor are right and wrong…… not or wrong but you’ve managed to make a statement that carries both values at the same time. A paradox.

Almost, but not quite right. Above I made the statement “There is an uncountable infinite number of real numbers between any two real numbers”. It is as simple as that. You use 0 and 1, but you could just as easily use .0000001 and .0000002. But it is not a paradox. How can it be. What exactly did Cantor say that is both right and wrong?

Now, why is the terminology countably infinite or uncountably infinite important. Because your example does not actually show uncountable infinite. Your example is countable. What does it mean to be countable? (or in your term finite infitinites, which does NOT make sense). It means, as you said above, you could fill every moment the rest of your life counting the numbers and if you lived forever you would never quite, BUT (and this is the important point) you would not leave any numbers out, you would not skip any. So they are countable. Its not a “finite infinity”.

Lets you your exampleof 1/2, 1/4, 2/4, 3/4, etc. That is countable. That is the set of rational numbers. I can arrange them in a matrix with only positive axis, lets list the numerators along the X axis and the denominators along the Y axis. Obviously the matrix goes to the right to infinity and goes up to infinity. If I start in the cell at 1/1 (the bottom left) and count moving up one, then moving diagaonal right-down to the x axis, then moving right one, then moving diaganol left-up to the right axis, then up one,then moving diagonal right-down, then right one…… I am counting every rationaly number, the ones greater and less than zero. So in this case, while there is an infinite number between 0 and 1, I can count them all (if I spend every moment of my life doing it and I never quite) without MISSING any of them. Hence that is a countably infinite set.

That is what Cantor shows, there are countable infinite sets and uncountable infinite sets, or stated as above “There is an uncountable infinite number of real numbers between any two real numbers”.

 

[tafan2]

NOTE: my illustration of counting rational numbers only counted positive rational numbers. One can modify it to count negative numbers also, but it is a little less easy to explain. But it suffices to show how to count all integers instead of just natural numbers (positive integers and zero). Do not start at negative infinity, start at zero, then count -1, then 1, then -2, then 2, then -3, then 3, etc.

 

[catholicray]

I definitely get lost on the logic when you start counting sets diagonally. You lose me there. But I’ll give it one more go as to what I am saying.

I am not aware if there is a math symbol for it so I’ll be using language
{0,1,2,3,4…∞} is less than and equal to {0,.5,1,1.5,2,2.5,3,3.5,4…∞}

Why? Because. LOL
Seriously I might as well start explaining things like children do at this point.
But I’ll explain beyond because. Why? Because that’s why.
So the above is true because ∞ = ∞ regardless of how you get there.

 

[tafan2]

I was trying to show how an two infinite sets, rational numbers and natural numbers are both countable even though one seemingly should be bigger than the other. There is a one-to-one mapping between the two sets. So in no way can you say their size is not equal. Pick any natural number, n, and I can show you precisely the n’th rational number. They are equal. The size of the set of natural numbers is NOT less than the size of the set of rational numbers There is no paradox.

 

[catholicray]

Ok in this I understand what you are saying but I disagree with your conclusion. That’s kind of funny. Okay this is my way of looking at it regardless of what someone is saying I have to believe. If I count to ∞ by 1/2’s
{.5,1,1.5,2,2.5,3…∞} I would count twice as many values than if I had counted {1,2,3,4,5…∞} therefore, I said that the counts are less than and equal to.

The reason you can map numbers as you say is because on a line of infinite numbers every number actually has the same value as every other number. Remember our {-3,-2,-1,0,1,2,3} number line? Well on an infinite number line the same number line would look like this {∞,∞,∞,∞,∞,∞,∞,∞} but where it really gets silly is that on the infinite number line {∞,∞,∞,∞,∞,∞,∞,∞} = {3,4,10,8,-4,7,-9,2} or any series of numbers you can imagine there. As we approach infinity our number line approaches paradoxical values.

Dividing by 0
So you know how when you divide by 0 on a graph and every number basically moves toward ∞ that’s because the value of 0 on those particular graphs is ∞
If we (here I go being crazy again) were to turn our graph on it’s side a 90 degree turn then what it would be telling us is that our 1 now has the value of ……… whatever value you want. Pick one.
But this is Chaos and to some degree gibberish.
Hence we don’t spend a lot of time dividing by 0

But wait there is more because also true is the fact that anything divided by 0 is 1. But this is whole different kind of animal than our innocent 1 on our 0,1,2,3 number line. This 1 is the full value of a real 1 an absolute value of 1 containing all of ∞ within it. Not a count of 1 but an absolute 1.

Off on a tangent a bit but I would say that you are able to map numbers because the value of number on an infinite number line is simply ∞ what you are in fact doing is mapping ∞ to ∞ Again utter gibberish an basically useless.

Regardless the way I see it is (if we decide to be reasonable):
On a number line {0,.5,1,1.5,2,2.5…∞} and a number line
{0,1,2,3,4,5…∞}
Every whole number would map to a number twice its size but inversely let’s check it out:

So 0,.5,1,1.5,2,2.5,3,3.5,4,4.5,5
we have to map all of this value
So 0,1,2,3,4,5,6,7,8,9,10

No if we are being reasonable then we would say that ∞ is the same number no matter how you count to it
Let’s be nice and just say that number is 100

Well {0,1,2,3,4,5…} Would have a count size of 100
Meanwhile {0,.5,1,1.5,2,2.5…} Would have a count size of 200 consequently this will be true every step of the way all the way to ∞
So this {0,1,2,3,4,5…} infinity is less than this {0,.5,1,1.5,2,2.5…} infinity
And ultimately it’s at the value of ∞ where all “rules” breakdown thus paradoxically they will also be the same size or equal to each other.

Sorry to write you a book but I am really enjoying our conversation so thanks for reading and being patient with me. I am open to changing my mind but thus far I just don’t see how I am actually making an error.

 

[Dovekin]

Let me try this a different way.

Between 0 and 1, we have B, the set of the inverses of the natural numbers.
B={b| b=1/n}
B is ∞, with 0<b<0.

Create C by removing 1/1 from B
C= B-1= ∞-1 0<c<1/2; C’={c’| c’=c+1/2} 1/2<c’<1
C = C’ = B-1= ∞
There are 2X ∞ between 0 and 1.

Create D by removing 1/2 from B.
D=B-2=∞. 0<d<1/3, 3X ∞ between 0 and 1

Continue to infinity, you get
Z= B-∞ = ∞-∞ = ∞. With ∞ X ∞ between 0 and 1

So 1= ∞ X ∞ by the logic you were using, a much larger number than ∞.
By the nature of ∞, this ∞ x ∞ = ∞ = ∞-∞.

The members of Z are called infinitesimals. They are all rational numbers. Z is countable because its members are defined to the natural numbers. There are more infinitesimals between 0 and 1, like 1/π,1/e, etc. Together, all the uncountable infinitesimals between 0 and 1 make up a larger ∞ than the rational ∞.

This is what Cantor was saying. There are countable infinities that seem like they should be ordered by size, ie 3x∞, 5x∞…Those infinities are all equal. But there are infinities that are larger than those infinities.

 

[catholicray]

Ahhhh now we’re getting somewhere. Your follow my logic. And you didn’t follow it all the way through…

What I am saying is far worse or better perhaps. I am saying that (∞ x ∞) is only the beginning of the value between 0 and 1. Your going to need to multiply what your saying infinitely more to see what I’m saying. Why?

Well first let me correct the picture. Between 0 and 1 exists an infinite infinity of value. Or to visualize this (∞x∞x∞x∞x∞x∞x∞x∞x…infinitely) Why indeed?

Because as I pointed out earlier every time we count to 1 we are actually counting (1∞)
Stay with me carefully now as I seem to have contradicted myself.

On a number line counting (1∞),(2∞),(3∞)…
∞ Reveals its real beastly nature
An infinite infinities or
(∞x∞x∞x∞x∞x∞x∞x∞x…infinitely) <— This! is the true value of infinity!!!

I’m going preach mode lol

∞ <— This!! is always a simplification of what is always true about infinity.

It’s easy my number line (1∞),(2∞),(3∞)… is a possibility and at the same time impossible.

We can count (1∞),(2∞),(3∞)… because we are ignoring! the full potential value of (1∞), which is fine so be it!

But… oh yes there’s a but…
If we observed the full value potential of (1∞) we could NOT count past 1

Paradox

The value available between 0 and 1 is like zooming into smaller worlds becoming larger … forever.