Infinity

[Dexter Sinister]

...now it is clear to me what you are trying to say. I will think about it.

I hope you enjoy thinking about such things, and I certainly assume you do or you wouldn't post what you do. Infinity's a slippery concept. I wonder if this will help: it's possible to represent any point in 2-D space (or 3-D, 4-D, etc.) by a single number just by changing the convention of how such things are specified. That's what I was doing with the interleaving of digits in the coordinates to reduce two numbers to one. The proof that Aleph One is infinitely greater than Aleph Null, or in other words that there are infinitely many more real numbers than integers, uses a similar tactic.

 

[JLM]

Yes, the integers are a subset of the real numbers. Who cares? Mathematicians.

Yep, they have a perfect right to their thoughts.

 

[SirJosephPorter]

Hello, Dexter, it is me again. I have been thinking about what you said, and it doesn’t add up. Now, if you got it from Gardner himself, I am not saying he is wrong. He may not be a mathematician, but I assume he took this material from mathematicians, who know their stuff. Nevertheless, there is a problem.

Let me summarize what you are saying. You are saying that there is a one to one correspondence between points in a square and points on a straight line. Show me point in the square (you are saying), and I will show you a point on the line, which will be unique to that point on the square.

And how do you arrive at that point? By alternating the digits. Thus, if I pick two points, a1a2a3 and b1b2b3, you will show me the point a1b1a2b2a3b3 as uniquely corresponding to those two points. Or if I pick a1a2a3 and b1b2, you will show me a1b1a2b2a3.

OK, now I am going to consider a square not 1 by 1, but 100 by 100. The principle is the same, of course, but we get away from fractions and decimals.. I pick points 24 ands 3. So, a1 = 2, a2 = 4, b1 = 3. So the number you show me is a1b1a2, or number 234.

Now let us look at the pair of numbers 2 and 34. a1 = 2, b1 = 3, b2 = 4. There is no a2, of course. The number is, a1b1b2, which is 234.

So both the pair of numbers, (24, 3) and (2, 34) give the same number 234.

What is wrong here? Is the method you describe flawed, or one to one correspondence does not exist?

As I said, I don’t want to say that Gardner is wrong, but there is some problem somewhere.

 

[Spade]

SJP, the set of real numbers is not a countable set.

 

[lone wolf]

Infinity: That place where the checkerboard sign is on the edge of the universe....

 

[Dexter Sinister]

...there is some problem somewhere.

Two problems, actually. First, as Spade indicated, the real numbers are not countable, which is, loosely speaking, just another way of saying you can't use them to count things with. Second, the principle is not the same, you've shifted the example to use integers, and the clue to the error is in your remark "there is no a2" in your second example. There must be an a2. All the numbers must have the same number of digits in them or you can't do the selection process as described, that's why I set it up as I did. You can't make the argument with integers because there aren't enough of them. It leaves out an infinite number of points in both the square and the line and you'll get, as you found, duplications.

The problem needs to be set up in a particular way in order to reason correctly about it, and integers aren't up to the task. The actual size of the line and the plane doesn't matter. You could, for instance, decide your units are 10,000 km long, call 10,000 km one Porter, and do all your calculations in terms of fractions of Porters. It's entirely arbitrary, you just have to treat them as having dimensions of a single unit of any size you like. They don't have to be equal on the line and the square either, you can make the square a megaPorter on a side and the line a microPorter long and use numbers that are fractions of megaPorters and microPorters, but it must be set up so that the numbers you deal with are the reals between 0 and 1 or the argument fails, as you saw.

 

[SirJosephPorter]

SJP, the set of real numbers is not a countable set.

I know that, Spade, but that is not what I am discussing with Dexter.

 

[SirJosephPorter]

it must be set up so that the numbers you deal with are the reals between 0 and 1 or the argument fails, as you saw.

Dexter, indeed the argument fails for numbers which are not between 0 and 1. Also, you requirement that the pair of numbers I choose must have the same number of digits sounds too restrictive.

So between 0 and 1, suppose I pick a pair of numbers, 0.2 and 0.43, and a pair of numbers 0.2 and 0.34. Wouldn’t both of them give the number .234 for the line?

OK, for your method to work, you are saying that both numbers must have the same numbers of digits. But then if I pick the numbers 0.2 and 0.43, how do you pick a number on the line which would correspond to that.

Doe that mean that the technique you described is incomplete, only partial and must be modified when the number of digits in the two numbers are not the same?

 

[Spade]

I know that, Spade, but that is not what I am discussing with Dexter.

You are discussing cardinality and mappings.
Before being too dismissive, you should investigate
1. How the cardinals Aleph null and Aleph 1 are related.
2. The continuum hypothesis
3. Well-ordering of the cardinals
How Aleph (n+1) is related to Aleph

As far as mappings are concerned, you can map the open interval (0, 1) onto the set of real numbers. It is just a matter of defining the function.

 

[Dexter Sinister]

Another thought, SJP: switch the numbers in your 100x100 square to reals, so your selections become (24.000..., 3.000...), and (2.000..., 34.000...). The duplication goes away, you get 2340000... in the first case and 2304000... in the second case. Then you have to decide where the decimal point goes in the results. I'd be inclined to put them where the first zero appears, 234.0000... and 23.040000..., but it's arbitrary. It's simpler to set it up so you're using only decimal fractions and there's only one obvious place to put the decimal point, at the beginning.

 

[Dexter Sinister]

So between 0 and 1, suppose I pick a pair of numbers, 0.2 and 0.43, and a pair of numbers 0.2 and 0.34. Wouldn’t both of them give the number .234 for the line?

I presume you mean (.24, .3) for the first pair, but in any event, no. Don't skip selecting a digit just because it's zero. 0.2 is actually 0.20000... for the purposes of this demonstration, so you get .2340 and .2304

 

[Dexter Sinister]

You are discussing cardinality and mappings.
Before being too dismissive, you should investigate...

A little slack,Spade. SJP's not a mathematician or he wouldn't be asking these questions. Those are likely unfamiliar concepts, I'm trying to explain things without jargon or mathematical formalism.

 

[Dexter Sinister]

This has been bugging me for days; I was sure I was missing something but couldn't dredge it up into the front of my mind. Finally came to me today: there is an Aleph Two. It's the number of curves that can be drawn in any 2-D or higher space. Through any point you can draw an infinite (Aleph One) number of curves.

To sum up:

Aleph Null: the number of integers, the number of rational numbers (i.e. numbers that can be expressed as a ratio of 2 integers), and the set that includes them both.

Aleph One: the number of real numbers, the number of points in a line, a plane, a cube, etc.

Aleph Two: the number of curves that can be drawn in a plane, a cube, etc.

Aleph Three and higher: nobody knows.

 

[SirJosephPorter]

This has been bugging me for days; I was sure I was missing something but couldn't dredge it up into the front of my mind. Finally came to me today: there is an Aleph Two. It's the number of curves that can be drawn in any 2-D or higher space. Through any point you can draw an infinite (Aleph One) number of curves.

To sum up:

Aleph Null: the number of integers, the number of rational numbers (i.e. numbers that can be expressed as a ratio of 2 integers), and the set that includes them both.

Aleph One: the number of real numbers, the number of points in a line, a plane, a cube, etc.

Aleph Two: the number of curves that can be drawn in a plane, a cube, etc.

Aleph Three and higher: nobody knows.

I don’t know Dexter, if integers are Aleph One (I agree) and real numbers are Aleph One (again, I agree, I assume you looked it up), then the next higher order is complex numbers.

Thus, there is ∞ to 1 correspondence between real numbers and integers. Also there is ∞ to 1 correspondence between complex numbers and real numbers (for each real number, there are infinite number of complex numbers). Also, it fits in nicely with their dimensional representations.

Integers (Aleph Null) can be represented with zero dimensions, i.e., a point (a set of points actually). Real numbers (Aleph One) can be represented with one dimension (a line), and complex numbers (Aleph Two?) can be represented with two dimensions, a plane.

I am not a mathematician. But if you say you looked it up and real numbers indeed are Aleph One, then I am convinced that complex numbers are Aleph Two.

Of course, this is not to contradict with what you are saying about number of curves, that also could be Aleph Two.

But again, I am not a mathematician. If mathematicians specifically say that complex numbers do not represent Aleph Two, I will take their word for it. But in the absence of them saying anything like that, I would consider complex numbers as Aleph Two (if you are saying that real numbers are Aleph One).

 

[Spade]

As you can plane-ly see, here lies Aleph 1. Enough infinity for a lifetime!

 

[SirJosephPorter]

Aleph One: the number of real numbers, the number of points in a line, a plane, a cube, etc.

Aleph Two: the number of curves that can be drawn in a plane, a cube, etc.

Dexter, this doesn’t make sense. Why would you say that number of points is a lower Aleph order than number of curves? If anything, there are more points than there are curves. Each curve passes through infinite number of points.

So why would you say that number of curves drawn in a plane is Aleph Two, but number of points is only Aleph One? Clearly there are more points than there are curves. If anything, it should be the other way round.

 

[Dexter Sinister]

Each curve passes through infinite number of points.

That's true, but think of it the other way around: through any single point on a curve there's an infinite number of possible curves. Imagine every point on a curve having an infinite number of curves drawn through it, then imagine the infinity of possible curves, with an infinite number of curves drawn through each point on all of them. Aleph Two is not Aleph One squared, as your complex numbers argument suggests, it's Aleph one times itself Aleph One times. And yes, I did look all this up, and I found another reference you might be able to find, George Gamow's little book One, Two, Three, Infinity. I saw it in Chapters today, in fact, it's back in print. Seeing it is what finally brought the memory to the surface.

 

[SirJosephPorter]

I have read one of his books, I don’t recall the name. In that he developed his hypothesis that all the heavy elements in the universe were formed within first 40 minutes after big bang (which of course we know now to be wrong).

Is that the same book, or is this a different one? If it is a different one, I will check it out.

 

[Dexter Sinister]

Is that the same book...

Honestly, I don't know. It's been over 30 years since I read it, I no longer have it (it was an ancient paperback that disintegrated). I didn't pick it off the shelf at Chapters and leaf through it today, I just saw the title on the spine, it jolted my memory about the Aleph numbers and I remembered that's where I first read about them. Isaac Asimov dealt with them in one of his thousands of essays too, but do you think I can find it? No way, he just wrote too much stuff.

I can, however, assure you that what I wrote about the Aleph numbers in post #73 is correct in terms of the way Georg Cantor first defined the methods of dealing with transfinite numbers.

 

[Niflmir]

I don’t know Dexter, if integers are Aleph One (I agree) and real numbers are Aleph One (again, I agree, I assume you looked it up), then the next higher order is complex numbers.

Complex numbers are essentially R^2 (there is a bijection between the two sets), the Hilbert curve (or one of many Peano curves) maps [0,1] onto [0,1]x[0,1] and [0,1] is isomorphic to R, therefore the cardinality of R and R^2 are the same.