to infinity, and then some...

[Fredrik]

No, this obviously wrong. x^a=x^b only if a = b, but n^m does not in general equal n*m

I think you've mistaken this for the relationship (xⁿ)^m = x^{n * m}. But the order of operations is different in this case, and the order of operations matters a lot.

Aw crap, I didn't think at all when I wrote the first paragraph in #47. I assumed that the number was (2^10)^118, but now when I think about it, it's not very likely that anyone would choose to express a number that way.

 

[69dodge]

The short answer is that you can't meaningfully define arithmetic operations on infinity.

No?

http://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic
http://en.wikipedia.org/wiki/Ordinal_arithmetic

Now, you CAN do arithmetic operations on variables and look at limits of those variables as you approach infinity.

That's a different sort of infinity than the kind that is used to measure the sizes of large sets.

I'm not sure exactly how they're related. But I don't think they're identical.

 

[Tim Thompson]

Infinite Infinities

No, what they were saying was that, in an infinite universe - i.e. not curved, just continuing out infinitely far - if you go far enough, the chances of coming across a planet identical to this one are 1 (in fact, there would be an infinite number of them) ...

I have seen this many times, but why should it be so? The space of all real numbers is as infinite as infinity gets, as far as I know. Yet there is one and only one of each number. Why should the infinity of an infinite universe be any different?

 

[Fredrik]

I have seen this many times, but why should it be so? The space of all real numbers is as infinite as infinity gets, as far as I know. Yet there is one and only one of each number. Why should the infinity of an infinite universe be any different?

There are good reasons to believe that a finite volume of space can only hold a finite amount of information. That makes this question very similar to the question of whether any given string of finitely many numbers between 0 and 9 can be found somewhere in the decimals of pi. But I don't think even that has been proved. (I hope someone will tell me if it has). If the universe is infinite, and the information content in every finite region is finite, then there's going to be repetitions somewhere, but I don't know if there really is a good reason to think that there's going to be a copy of this solar system out there somewhere. (I strongly doubt it).

 

[Ambrosia]

You couldn't write it. you would have to write more digits than there are atoms in the visible universe.

OK - is there any way of visualising just how big this number is?

I thought (presumably mistakenly) that 2¹⁰¹¹⁸ was

(10 multiplied by itself 118 times) then 2 multiplied by itself that many times again.

Up until yesterday I didn't know you could have numbers that have powers ontop of their powers.

If you do the grain of rice on a chessboard thing whereby you double the number of grains of rice effectively 64 times, you wind up with about 461 billion tonnes of rice.

Thats 2⁶³ (!!)

this is 2^{118 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 0}
(spaces added so it doesn't wreck the pagination)

My brain is melting again


ETA: 461 billion tonnes of rice would cover the surface area of India ~1m deep in rice. Thats what I mean about trying to visualise how big this number is. I get that "more than the atoms in the known universe" is reaaaaaaaaaally big, but I can't relate to how big. If anyone who understands this stuff is really bored sometime soon and can work out an illustration like say "this much dust would cover an area the size of the orbit of pluto 7 trillion miles deep" (arbitrary numbers pulled out of thin air) that would be superb.

 

[W.D.Clinger]

Look, I know this has already been answered, but I still wonder about the business of

Infinity - infinity not equal to 0

Subtraction of infinite cardinals is not well defined.

It just feels like when I say 6-6=0 I am dealing with one thing and when I say "I have a hotel with infinite rooms and one guest stays behind etc etc" I am dealing with a different kind of thing. So, whereas I find the other arguments about the set of all numbers and the set of even numbers being the same size of infinity subjectively acceptable, I don't see the argument about subtraction as being acceptable to me in the same way. Can someone tell it to me a different way?

Your misgivings are well founded (no pun intended, and those who understand the potential pun are among the few who will understand this post).

Suppose I have a bag containing an infinite set of balls, labelled 0, 1, 2, et cetera. I reach into the bag and remove two balls, choose one of the balls, and put the chosen ball back into the bag. I repeat that operation ad infinitum.

That takes infinitely long, but how many balls are in the bag when I'm done?

The answer is not well defined. It can be any finite cardinal you choose, or the first infinite cardinal ([latex]\aleph_0[/latex]).

You want proof? Okay. I withdraw balls 0 and 1, and put ball 0 back. I withdraw balls 1 and 2, and put ball 1 back. I withdraw balls 2 and 3, and put ball 2 back. I withdraw balls 3 and 4, and put ball 3 back. Et cetera. When I'm "done", every ball has been withdrawn, put back, and never withdrawn again.

Ah, but how could we end up with exactly 3 balls in the bag? I withdraw balls 0 and 1, and put ball 1 back. I withdraw balls 1 and 2, and put ball 2 back. I withdraw balls 2 and 3, and put ball 2 back. From then on I proceed as follows. I withdraw balls 3 and 4, and put ball 4 back. I withdraw balls 4 and 5, and put ball 5 back. I withdraw balls 5 and 6, and put ball 6 back. When I'm "done", the bag will contain balls 0, 1, and 2. Balls 3, 4, 5, 6, 7, 8, et cetera will have been removed without being put back.

And similarly for any other finite cardinal.

 

[69dodge]

I thought (presumably mistakenly) that 2¹⁰¹¹⁸ was

(10 multiplied by itself 118 times) then 2 multiplied by itself that many times again.

You're mistaken about being mistaken.

this is 2^{118 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 0}

That has the right number of zeros, but the 118 should just be 1.

 

[realpaladin]

In this thread there's also a lot of fun about infinity and cardinality.

Am going to invite a few of them over

 

[69dodge]

Oh no. Not Doron.

(He is a very skillful troll, and likely will completely ruin the thread. I recommend that no one reply to him if he posts here.)

 

[Cavemonster]

Right, I should have said an infinitely dividable ball can be split into 5 pieces, have two of the pieces rotated, and the pieces can be reassembled into 2 balls the same size as the original.

I don't think a non-continuous collection of points is a very good 'piece' of something, but it's really not an area where I should try and use my intuition about volume.

<pedant>
You can do this for a sphere mathematically, not a ball. A ball is a real world object with granularity.
</pedant>

 

[jsfisher]

In this thread there's also a lot of fun about infinity and cardinality.

Am going to invite a few of them over

I can't believe you invited doron to this thread.


That aside, is this thread now about ready for the fact there are more real numbers between 0 and 1 then there are integers?

 

[Molinaro]

Unless I'm also confused, while both sets of numbers might be infinite, the first infinite is twice as big as the second infinite. (Not all values of infinite are the same.)

You are confused on the 1st point, and correct on the point in parenthesis. Those two infinite sets are equal because you can setup a 1 to 1 relationship between their members, as already shown above.

Here is an example of 3 progressively larger infinites:

1) the infinite number integers
2) the infinite number of points in a plane
3) the infinite set of lines generated by considering for each point in a plane, the infinite number of lines of all possible shapes passing through that point

 

[jsfisher]

You are confused on the 1st point, and correct on the point in parenthesis. Those two infinite sets are equal because you can setup a 1 to 1 relationship between their members, as already shown above.

Here is an example of 3 progressively larger infinites:

1) the infinite number integers
2) the infinite number of points in a plane
3) the infinite set of lines generated by considering for each point in a plane, the infinite number of lines passing through that point

Hmmm, I'm going to have to think about that one. I know the number of curves in a plane exceeds the number of points, but I had thought points and lines were the same.

ETA: In fact, I am fairly certain of it. Any line can be identified by two points. Each point can be identified by a pair of real numbers. I can encode all four numbers into a single real number by simply interleaving the digits.

 

[Cavemonster]

Hmmm, I'm going to have to think about that one. I know the number of curves in a plane exceeds the number of points, but I had thought points and lines were the same.

ETA: In fact, I am fairly certain of it. Any line can be identified by two points. Each point can be identified by a pair of real numbers. I can encode all four numbers into a single real number by simply interleaving the digits.

That would only hold for lines within that plane, I think.

I believe Molinaro's set includes all lines defined by each of those points and points outside of the plane.

 

[Molinaro]

Actually I should have used the word curves, not lines. I meant lines as in take a pencil and draw all possible squiggly or straight lines that pass through the point in question. For any two points you choose in the plane, they will each have an infinite many lines that don't pass through the other. The total number of all curves, is the next larger infinite above the size of the set of reals.

 

[jsfisher]

Actually I should have used the word curves, not lines. I meant lines as in take a pencil and draw all possible squiggly or straight lines that pass through the point in question. For any two points you choose in the plane, they will each have an infinite many lines that don't pass through the other. The total number of all curves, is the next larger infinite above the size of the set of reals.

Then we agree completely.

That would only hold for lines within that plane, I think.

If you consider lines in 3-space instead of just the plain, then you need three numbers for the coordinates of each of two points to identify the line. You have to interleave 6 numbers into 1 instead of 4 into 1, but it is still the same level of infinity.

 

[Brian-M]

Your way of thinking about "densities" is not very fruitful, as not all sets lend themselves to be pictured on a line (or an area, or ...).

The standard way to compare infinite sets is by establishing functions between them.

An (infinite) set A has smaller or equal cardinality ("size") than a set B, if you can make an injective function (aka one-to-one function)
f: A -> B
Injective means that:
f(x) != f if x != y

Thanks for taking the time to respond, but I'm afraid it's lost on me (and the amount of time and effort it would take me to research this in order to understand what you're getting is too much just for the sake of discussion--maybe I'll look it up some other time).

 

[Brian-M]

No. For example, there's no way to do it for the integers and the real numbers. There will always be lots of real numbers left over, however we do it. So, we say that there are more real numbers than integers.

You can make the same kind of 1:1 correspondence between integers and any set of numbers with n decimal places by multiplying the set by 10ⁿ. As real numbers are numbers with infinite decimal places, couldn't you make a 1:1 correspondence by multiplying the set of real numbers by 10^infinite?

 

[Vorpal]

<pedant>
You can do this for a sphere mathematically, not a ball. A ball is a real world object with granularity.
</pedant>

<pedant>A ball is a mathematical object defined as the set of all points within some fixed distance from a given center (≤ if closed, < if open). A sphere is the set of points that are exactly some fixed distance from a given center.</pedant>

You can make the same kind of 1:1 correspondence between integers and any set of numbers with n decimal places by multiplying the set by 10ⁿ. As real numbers are numbers with infinite decimal places, couldn't you make a 1:1 correspondence by multiplying the set of real numbers by 10^infinite?

But you don't get integers in that case, just infinite strings of digits. And there are indeed as many infinite strings of digits as real numbers.

 

[Hampster]

Thanks to Xeno (or Zeno) Infinity was a forbidden topic in Math for many, many years. Newton and Leibnitz invented Calculus, but this dealt with infinity, so many people thought of Calculus as some sort of "trick" or "gimmick". Yes, it produced results, but after the results were obtained, other methods were used to "prove" the results that Calculus gave.

It was not until Cantor, and an understanding of infinity, that Calculus became trusted. And Cantor, the first human to understand this concept, spent much of his life in a mental institution.

--Dave