Infinity

[Gavin Chipper]

Maybe I'm a bit like Kirk Bevins is with induction but the idea that infinities come in different sizes is barking to me, so I'm interested in what you all think of it.

Here's a link: http://en.wikipedia.org/wiki/Cantor%27s ... l_argument

[Charlie Reams]

the idea that infinities come in different sizes is barking to me

The 19th Century called, they want their understanding of mathematics back.

[Clare Sudbery]

It makes perfect sense to me. This kind of stuff was my favourite bit of studying maths at degree level. I love it, because it is so counter-intuitive and mad but makes sense as soon as you go into it.

The easiest way (for me) of thinking about it is to think of the different infinite number sets. I'm a bit rusty though, so forgive me if I get some of the terminology wrong.

There are the integers. These are the numbers you can count. So: 1, 2, 3 etc. These go on forever, right? There is no last number. You can always add one. So this is an infinite set. But then there are the rational numbers, which include the fractions. 1/2, 1/4 etc. There are an infinite number of fractions between 0 and 1 alone - because you start with 1/2 and keep going to 1/n, and there will always be 1/(n+1). And then you can start all over again with 1.25, 1.5 etc. So there are clearly MORE rational numbers than there are integers. Name any integer you like - eg n - and I will show you another infinite number of rational numbers (1/n, 1 + 1/n ... n + 1/n... and so on). There are definitely more of them. So although the size of the set of rational numbers is infinity, and the size of the set of integers is also infinity... it's clear that the first infinity is bigger than the second one. Which does indeed sound like nonsense, but has all sorts of practical applications (which sadly I forget) and lead you on to a whole load of other mind-blowing stuff.

I love the kind of maths which you can make sense of by thinking logically and following step A with step B etc, but when you step back and try and visualise it, or grasp it intuitively, it blows your brain apart. Like the concept of +4 dimensions. Or the universe going on forever. Or how small you are compared with a galaxy. Or how big you are compared to an atom. Or how the distance between the various components of a particle is proportionally as large as the distance between planets in our galaxy. Or what came before the big bang. Or entangled particles*. Or how you can cease to exist when you die. It makes logical sense but it also has a kind of magic. And of course religion feeds on the doubts and confusions caused by the difficulty in grasping these truths, like a large and voracious feedy thing. With big teeth.

*Tenuous claim to fame: That's my dad, that is.

[Paul Howe]

So there are clearly MORE rational numbers than there are integers.

I'm afraid this isn't the case. The rationals can in fact be put into a 1-1 correspondence with the integers, it's quite a cute little problem to work out how its done.

[Michael Wallace]

This thread makes me want to cry already.

[Gavin Chipper]

So there are clearly MORE rational numbers than there are integers.

I'm afraid this isn't the case. The rationals can in fact be put into a 1-1 correspondence with the integers, it's quite a cute little problem to work out how its done.

I was going to say that as well. I'll post my specific "Kirks" (objections) to the idea of different sizes in a bit. Maybe today if I get round to it.

[Charlie Reams]

What Paul said. It's the real numbers that outnumber (ha!) the integers.

[Junaid Mubeen]

So there are clearly MORE rational numbers than there are integers.

I'm afraid this isn't the case. The rationals can in fact be put into a 1-1 correspondence with the integers, it's quite a cute little problem to work out how its done.

Indeed...I think you've missed the point here, Clare. If you'd said irrational numbers (those which are not rational) you'd be correct, but as Paul says the rational numbers can be counted. So in terms of this 'level of infinity', or 'cardinality' to give it the proper name, rationals = integers < irrationals = reals. This last bit is a horrible abuse of notation but hopefully you get the point.

In fact, the cardinality of the reals is strictly less than the cardinality of its power set (which is the collection of all subsets of the reals).

A mars bar to anyone who can find a set whose cardinality lies strictly between these two sets

Edit: I realise this post may have caused some confusion. Just to clarify, I meant Paul Howe and not Paul. My apologies.

[Clare Sudbery]

So there are clearly MORE rational numbers than there are integers.

I'm afraid this isn't the case. The rationals can in fact be put into a 1-1 correspondence with the integers, it's quite a cute little problem to work out how its done.

Indeed...I think you've missed the point here, Clare. If you'd said irrational numbers (those which are not rational) you'd be correct, but as Paul says the rational numbers can be counted. So in terms of this 'level of infinity', or 'cardinality' to give it the proper name, rationals = integers < irrationals = reals. This last bit is a horrible abuse of notation but hopefully you get the point.

Haha, brilliant. I told you I was rusty. You're right, of course. It's coming back to me now (I graduated 18 years ago and have taken a lot of drugs since then). So, counter-intuitiveness on top of counter-intuitiveness.

That's another thing I like: A good proof to something counter-intuitive. Well, not so other I guess, just more of the above.

Peano's Axioms were amongst my favourite things. And Godel's Incompleteness Theorem. VFSMB to anyone who can explain them. When I'm 100 years old and have run out of things to do, I plan to revisit all of this stuff. Life's just too short.

[Paul Howe]

Edit: I realise this post may have caused some confusion. Just to clarify, I meant Paul Howe and not Paul. My apologies.

I just cannot wait until another Junaid turns up on here. There's already been two with the same forename AND surname, so its only a matter of time.

[Gavin Chipper]

OK, here's a summary of my concerns over the matter.

I understand the diagonal proof which shows that you can't put, for example, the real numbers into one to one correspondence with the integers. However, while such a method clearly distinguishes between sizes in finite sets, I think that this is possibly stretching things beyond their usefulness/relevance.

Suppose it's not sets that I want to put in order of size, but lists. Lists that have an order. So your first list might be the integers in ascending order starting from 1. And your second list might be the integers, starting with the multples of 17, and then doing the rest in ascending order. But then we find that we can't put the lists into one to one correspondence with each other! The one starting with the multiples of 17 always has numbers left over so there are more of them even though we have exactly the same numbers in the lists!

Obviously the above is slightly taking the piss, but that doesn't matter. My point is that what works for sets works for lists. What makes sets the special thing that has a size that can be measured in this way and not a list?

My other point from this is that while the integers can't be put into one to one correspondence with the real numbers, is this not really because the real numbers are awkward and inaccessible rather than because there are more of them? They are awkward because you can't put them in a decent order. The integers only get accused of being small because for each one of them there is an obvious next number. You know, you could have the biggest set in the whole world but because each member of the set has an obvious sequel, it is discriminated against!

It's also worth noting that while there are "more" real numbers than rational numbers, that doesn't stop the fact that for any two real numbers you can find a rational number between them.

Also, I've heard it said that for any infinity, you can make a bigger one by doing 2 to the power of it. But what happens if you take the log of aleph null? Or is it a dead end that way?

So we've got the integers (all aleph null of them) and to 2 to the power of it. What do we get? Here's a list of the integers:

2^ZERO, 2^ONE, 3, 2^TWO, 5, 6, 7, 2^THREE, 9, 10, 11, 12, 13, 14, 15, 2^FOUR, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 2^FIVE, 33 and so on.

I've written some of the numbers in capitals but it's all the same. You will also notice that the integers are written out a second time, in capital letters. Now as this list continues forever, the numbers in capital letters will obviously "approach" aleph null. So the main number list must approach 2 to the power of aleph null. But it also, being a list of integers, approaches aleph null, so 2 to the power of aleph null must be aleph null!

Maybe I'm errroneously trying to claim that something that works for finite numbers (that the numbers in the main list are 2 to the power of the numbers in capitals) also works for the infinite. But if so, the whole one to one correspondence thing was working under the assumption that if it's good for the finite it's good for the infinite.

[Charlie Reams]

My point is that what works for sets works for lists. What makes sets the special thing that has a size that can be measured in this way and not a list?

Nothing. In fact it's the very fact that you can put the integers into a list that distinguishes them from the reals. You can't turn the reals into a list, because there's just no way to traverse them all in linear order.

Also, I've heard it said that for any infinity, you can make a bigger one by doing 2 to the power of it. But what happens if you take the log of aleph null? Or is it a dead end that way?

So we've got the integers (all aleph null of them) and to 2 to the power of it. What do we get? Here's a list of the integers:

2^ZERO, 2^ONE, 3, 2^TWO, 5, 6, 7, 2^THREE, 9, 10, 11, 12, 13, 14, 15, 2^FOUR, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 2^FIVE, 33 and so on.

I think you've misunderstood, so I'll try to clarify but with the firm caveat that I might also be wrong and fully expect to be jumped on by someone better qualified. (I'm going to write A for aleph null here.) You can define a larger cardinal than A by taking 2^A, which also turns out to be the size of the power set of integers (i.e. the number of sets of integers.) This is Aleph 1 and is "larger" than A. You can repeat the same trick to define a (countably) infinite hierarchy of ever larger infinities.

[Gavin Chipper]

Nothing. In fact it's the very fact that you can put the integers into a list that distinguishes them from the reals. You can't turn the reals into a list, because there's just no way to traverse them all in linear order.

OK, but you also can't put the integers-while-ordered-in-a-specific-way into a list and that's what distinguishes them from the reals-in-no-particular-order.

I think you've misunderstood, so I'll try to clarify but with the firm caveat that I might also be wrong and fully expect to be jumped on by someone better qualified. (I'm going to write A for aleph null here.) You can define a larger cardinal than A by taking 2^A, which also turns out to be the size of the power set of integers (i.e. the number of finite sets of integers.) This is Aleph 1 and is "larger" than A. You can repeat the same trick to define a (countably) infinite hierarchy of ever larger infinities.

Well, I dunno. You've told me that 2^A is larger than A but I'm not sure you've explained why. But I'm going to bed now in the hope that this thread will have loads of replies when I next read it (probably tomorrow evening).

[Charlie Reams]

Nothing. In fact it's the very fact that you can put the integers into a list that distinguishes them from the reals. You can't turn the reals into a list, because there's just no way to traverse them all in linear order.

OK, but you also can't put the integers-while-ordered-in-a-specific-way into a list and that's what distinguishes them from the reals-in-no-particular-order.

Why can't you put the integers-in-a-specific-order into a list?

Well, I dunno. You've told me that 2^A is larger than A but I'm not sure you've explained why. But I'm going to bed now in the hope that this thread will have loads of replies when I next read it (probably tomorrow evening).

You're right, I didn't even attempt to explain that. One way to think about it is to say that if you have a set S with size n, the number of subsets of S is 2^n, because in each subset you either include or exclude each element, so you make a two-way choice n times. (Not sure if I expressed that well, but if you think of finite sets then it's easy to see.) In the case where S is the integers, you know that n = A, so the size of P(S) is 2^A. The question is then, is 2^A bigger than A? The answer is yes, because there's a 1-to-1 mapping of sets of integers on to real numbers, and we already established that sizeof(set of real numbers) is greater than A.

Devising the mapping between integer sets and real numbers is left as an exercise for the reader

[Chris Corby]

.................I graduated 18 years ago and have taken a lot of drugs since then

Well, that explains Monday.

And as far as Mathematics is concerned, a minute never passes because when half a minute is gone there is still a half to go and when half of that passes there is still a quarter to go... and so on...... to infinity and beyond.

[Phil Reynolds]

Peano's Axioms were amongst my favourite things. And Godel's Incompleteness Theorem. VFSMB to anyone who can explain them.

I could have done the latter for you on demand a year ago - in Breaking the Code I had a three-page speech in which I explained Godel's Theorem among other things.

[Clare Sudbery]

Peano's Axioms were amongst my favourite things. And Godel's Incompleteness Theorem. VFSMB to anyone who can explain them.

I could have done the latter for you on demand a year ago - in Breaking the Code I had a three-page speech in which I explained Godel's Theorem among other things.

Acting and maths together! What a lovely gig. I'm jealous.

[Charlie Reams]

.................I graduated 18 years ago and have taken a lot of drugs since then

Well, that explains Monday.

And as far as Mathematics is concerned, a minute never passes because when half a minute is gone there is still a half to go and when half of that passes there is still a quarter to go... and so on...... to infinity and beyond.

Ahhh, Xeno's paradox, my old friend.

[Michael Wallace]

And as far as Mathematics is concerned, a minute never passes because when half a minute is gone there is still a half to go and when half of that passes there is still a quarter to go... and so on...... to infinity and beyond.

Ahhh, Xeno's paradox, my old friend.

So if you cut a sandwich in half, and then cut those halves in half...

[Phil Reynolds]

Acting and maths together! What a lovely gig. I'm jealous.

I've also done acting and astrophysics: in Humble Boy I had to explain superstring theory, although (it must be said) with rather less rigour.

[Chris Corby]

.................I graduated 18 years ago and have taken a lot of drugs since then

Well, that explains Monday.

And as far as Mathematics is concerned, a minute never passes because when half a minute is gone there is still a half to go and when half of that passes there is still a quarter to go... and so on...... to infinity and beyond.

Ahhh, Xeno's paradox, my old friend.

Sorry mate, but you've made yourself look a bit of an idiot here. It's actually Toy Story.

[Jon Corby]

... and so on...... to infinity and beyond.

Ahhh, Xeno's paradox, my old friend.

Sorry mate, but you've made yourself look a bit of an idiot here. It's actually Toy Story.

[Kai Laddiman]

Infinity starts with a 5.

[Charlie Reams]

Sorry mate, but you've made yourself look a bit of an idiot here. It's actually Toy Story.

I've taken a lot of drugs since I last watched a Disney movie.

[Gavin Chipper]

Why can't you put the integers-in-a-specific-order into a list?

It's not that you can't put the integers-in-any-specific-order into a list but that you can't put the integers-in-a-specific-specific-order into a list. The example I gave earlier - start with the multiples of 17. If you write the integers in normal ascending order, then you can pick any integer you want and you will find it in your list in a finite amount of time if you go down far enough. If you start with the multiples of 17, then you will simply never reach 6, for example, however far you go down.

You're right, I didn't even attempt to explain that. One way to think about it is to say that if you have a set S with size n, the number of subsets of S is 2^n, because in each subset you either include or exclude each element, so you make a two-way choice n times. (Not sure if I expressed that well, but if you think of finite sets then it's easy to see.) In the case where S is the integers, you know that n = A, so the size of P(S) is 2^A. The question is then, is 2^A bigger than A? The answer is yes, because there's a 1-to-1 mapping of sets of integers on to real numbers, and we already established that sizeof(set of real numbers) is greater than A.

That does make sense (I presume we're meant to take it as read that there is a one-to-one mapping of sets of integers on to real numbers). But I would still dispute that the one-to-one correspondence thing is a determiner of size for infinite numbers, but more to do with the inaccessibility/awkwardness of the numbers, which stops you from listing them.

And my "proof" that 2^A = A still seems to work to me.

Devising the mapping between integer sets and real numbers is left as an exercise for the reader

I think I've seen how to do that before so I'll leave it to someone else.

[Paul Howe]

And my "proof" that 2^A = A still seems to work to me.

Assuming you're referring to this.....

Also, I've heard it said that for any infinity, you can make a bigger one by doing 2 to the power of it. But what happens if you take the log of aleph null? Or is it a dead end that way?

So we've got the integers (all aleph null of them) and to 2 to the power of it. What do we get? Here's a list of the integers:

2^ZERO, 2^ONE, 3, 2^TWO, 5, 6, 7, 2^THREE, 9, 10, 11, 12, 13, 14, 15, 2^FOUR, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 2^FIVE, 33 and so on.

I've written some of the numbers in capitals but it's all the same. You will also notice that the integers are written out a second time, in capital letters. Now as this list continues forever, the numbers in capital letters will obviously "approach" aleph null. So the main number list must approach 2 to the power of aleph null. But it also, being a list of integers, approaches aleph null, so 2 to the power of aleph null must be aleph null!

But how can aleph-null, an infinite cardinal number, appear in a list of finite numbers? This is where your argument falls down.

[Gavin Chipper]

But how can aleph-null, an infinite cardinal number, appear in a list of finite numbers? This is where your argument falls down.

It doesn't appear in the list but I don't think that changes the point. In the list written in captials it never reaches aleph null but there are aleph null numbers in the list. Likewise, in the other number list, it never reaches 2 ^ aleph null so can't it still be said that there are 2 ^ aleph null numbers in the list?

In the same way that 0.1 + 0.01 + 0.001 etc approaches 1/9 my number lists approach aleph null and 2 ^ aleph null.

[Rosemary Roberts]

So if you cut a sandwich in half, and then cut those halves in half...

Crumbs!!

[Paul Howe]

But how can aleph-null, an infinite cardinal number, appear in a list of finite numbers? This is where your argument falls down.

It doesn't appear in the list but I don't think that changes the point. In the list written in captials it never reaches aleph null but there are aleph null numbers in the list. Likewise, in the other number list, it never reaches 2 ^ aleph null so can't it still be said that there are 2 ^ aleph null numbers in the list?

In the same way that 0.1 + 0.01 + 0.001 etc approaches 1/9 my number lists approach aleph null and 2 ^ aleph null.

Not really, in order for a sequence to approach a limit (in this case, A or 2^A), you need some sense of being able to get arbitrarily close to the limit by incorporating more and more terms into the sequence. As c-z = c for any infinite cardinal c and integer z, you're not actually getting any closer at all.

[Junaid Mubeen]

Hm...Gavin, what do you mean by 2^A ? It is NOT all things of the form 2^a, for elements a of A. Rather, it is the power set of A, described above somewhere. With the first definition, 2^A would indeed have the same cardinality as A, but the power set is 'bigger'.

[Paul Howe]

He's using A as shorthand for aleph-null, not to denote a set.

[Gavin Chipper]

He's using A as shorthand for aleph-null, not to denote a set.

Yeah and that's Charlie's fault. He started it.

OK, I take your point about not getting any closer and not "approaching" but I'm still not convinced by all of this. I would still say that while you can "count" to aleph null (you can't actually reach it but it's deemed "countable" in some sense anyway) you can also "count" to 2 ^ aleph null in the same way.

And what does happen if you take the log of aleph null?

And my list of integers starting with the multiples of 17 is still not countable.

[Charlie Reams]

Why can't you put the integers-in-a-specific-order into a list?

It's not that you can't put the integers-in-any-specific-order into a list but that you can't put the integers-in-a-specific-specific-order into a list. The example I gave earlier - start with the multiples of 17. If you write the integers in normal ascending order, then you can pick any integer you want and you will find it in your list in a finite amount of time if you go down far enough. If you start with the multiples of 17, then you will simply never reach 6, for example, however far you go down.

That's true, it is possible to come up with an order which doesn't traverse all the numbers. But that's irrelevant. The point is that if you can find any way of pairing up the elements in each set then they must have equal cardinality.

What you're saying is equivalent to saying "Here's a failed attempt to prove X, therefore X is false."

[Gavin Chipper]

That's true, it is possible to come up with an order which doesn't traverse all the numbers. But that's irrelevant. The point is that if you can find any way of pairing up the elements in each set then they must have equal cardinality.

What you're saying is equivalent to saying "Here's a failed attempt to prove X, therefore X is false."

Yes, I understand this point. But my point was that why is it sets, as opposed to lists, that have sizes anyway? Maybe it is possible to change the size of something by changing its order.

[Charlie Reams]

Yes, I understand this point. But my point was that why is it sets, as opposed to lists, that have sizes anyway? Maybe it is possible to change the size of something by changing its order.

You could define a notion of size for lists, but you can't assume it'll behave the same as size for sets, which is the crux of your argument-by-analogy.

[Gavin Chipper]

Yes, I understand this point. But my point was that why is it sets, as opposed to lists, that have sizes anyway? Maybe it is possible to change the size of something by changing its order.

You could define a notion of size for lists, but you can't assume it'll behave the same as size for sets, which is the crux of your argument-by-analogy.

Well, it seems that people have simply decided that the one-to-one correspondence thing is the measure of size for infinite sets. I've seen nothing more than this. So I can make the same claim for lists.

It's not as if someone has found that the real numbers can't be put into one to one correspondence with the integers and then found in addition to this that this means that there are more real numbers. One to one correspondence is the beginning and the end of it. Likewise my integer list starting with the multiples of 17 v integers ordered normally. They can't be put into one to one correspondence so I can simply say that one list is bigger.

[Charlie Reams]

Well, it seems that people have simply decided that the one-to-one correspondence thing is the measure of size for infinite sets. I've seen nothing more than this. So I can make the same claim for lists.

The definition of cardinality (which you can take to mean size if you like) is "two sets have the same cardinality if I can pair each item in one with an item in the other". It seems a very natural definition, clearly agrees with our intuition for finite sets, and it obeys the definition of an equivalence operator. But it's just a definition. You're free to define some other idea and see whether it gets you anywhere.

It's not as if someone has found that the real numbers can't be put into one to one correspondence with the integers and then found in addition to this that this means that there are more real numbers. One to one correspondence is the beginning and the end of it.

Yes, that's the definition.

Likewise my integer list starting with the multiples of 17 v integers ordered normally. They can't be put into one to one correspondence so I can simply say that one list is bigger.

Sure, but you're not disagreeing with any established mathematics.

[Gavin Chipper]

Sure, but you're not disagreeing with any established mathematics.

Am I not? Wouldn't mathematicians disagree that one of those lists is bigger than the other? OK - I suppose there isn't an established area of maths about "infinite lists".

So, it comes down to how we define "bigger"? What we're simply talking about is cardinality of sets and it also happens to fit some people's intuitive idea of bigness of numbers.

And finally, what happens if you take the log of aleph null?

[Rosemary Roberts]

You could define a notion of size for lists, but you can't assume it'll behave the same as size for sets, which is the crux of your argument-by-analogy.

Surely any list is also a countable set? So anything you can say about sets must hold true for a list. But you cannot conversely apply the properties of a list to a non-countable set. Is the notion of a list any help at all?

what happens if you take the log of aleph null?

The short answer is that you don't - the question is meaningless.

[Charlie Reams]

what happens if you take the log of aleph null?

The short answer is that you don't - the question is meaningless.

Are you sure about that? Presumably if you take the log (base 2) of aleph one then you get aleph null, so it seems reasonable to ask what log(aleph_null) is, even if the answer is "there's no such number."

[Andrew Feist]

Sure, but you're not disagreeing with any established mathematics.

Am I not? Wouldn't mathematicians disagree that one of those lists is bigger than the other? OK - I suppose there isn't an established area of maths about "infinite lists".

So, it comes down to how we define "bigger"? What we're simply talking about is cardinality of sets and it also happens to fit some people's intuitive idea of bigness of numbers.

And finally, what happens if you take the log of aleph null?

I think you might be treading on some established areas of maths -- you could probably write this notion you have of a list in the language of partially ordered sets. 17 would be a bottom element (as 17 < x for every x), 17x < 1 for every natural number x (since they would precede it). In fact you would have a total order, since every pair of numbers would be comparable. But 1 would be a special kind of element, as the set of all x with x < 1 doesn't have an upper bound. I'm pretty sure there's a name for that sort of thing, but I'd have to look it up 'cause I don't remember.

[Rosemary Roberts]

what happens if you take the log of aleph null?

The short answer is that you don't - the question is meaningless.

Are you sure about that? Presumably if you take the log (base 2) of aleph one then you get aleph null, so it seems reasonable to ask what log(aleph_null) is, even if the answer is "there's no such number."

Not "sure", exactly - it's too long ago - but one of the first and most painful things I learned at college is that "presumably" doesn't cut it. It's not so much that there is no such number but that the equation is insolvable. The log of any integer is defined, but not in a way that could take a limit. It would also not be any use, because - so far as I recall - adding transfinite numbers doesn't change them.

Obviously the transfinite numbers crowd can define it, if they choose, but I'm not aware that anybody every has.

[Gavin Chipper]

what happens if you take the log of aleph null?

The short answer is that you don't - the question is meaningless.

Are you sure about that? Presumably if you take the log (base 2) of aleph one then you get aleph null, so it seems reasonable to ask what log(aleph_null) is, even if the answer is "there's no such number."

Couldn't such a number be "invented"? Presumably a while back the answer to "What's the square root of minus one" was "There's no such number". We could define this to be an infinity that is smaller than the number of integers even if there is no set that actually contains this number of elements. Obviously we can take the log of this new infinity as well and so on.

[Rosemary Roberts]

Couldn't such a number be "invented"? Presumably a while back the answer to "What's the square root of minus one" was "There's no such number". We could define this to be an infinity that is smaller than the number of integers even if there is no set that actually contains this number of elements. Obviously we can take the log of this new infinity as well and so on.

Sure, you can define it how you like, preferably so as to be consistent with the rest of the theory. Which is naturally the hard part. But be my guest.

[Gavin Chipper]

Horizon at 9:00 is about infinity. They didn't ask me on to talk though.

[Gavin Chipper]

Anyone watch it? I thought it was quite interesting. One point they made was that if the universe was infinitely big, everything that's possible would definitely happen (an infinite number of times as well). So this would include a monkey typing out the entire works of Shakespeare. But what they neglected to mention was that an infinite space doesn't necessarily mean infinite matter. In fact my intuitive view of the universe while growing up was that it had infinite space but not infinite matter.

Also I'm not sure if this is a sensible analogy or not, but against someone like me saying that you can't get bigger than infinity because you can just go on and on forever, you could compare an infinite plane (you just go on and on forever!) with an infinite volume, which presumably most people would call bigger.

[Charlie Reams]

Also I'm not sure if this is a sensible analogy or not, but against someone like me saying that you can't get bigger than infinity because you can just go on and on forever, you could compare an infinite plane (you just go on and on forever!) with an infinite volume, which presumably most people would call bigger.

Maybe most people would say that, but the two sets have the same cardinality (i.e. "size").

[Gavin Chipper]

Also I'm not sure if this is a sensible analogy or not, but against someone like me saying that you can't get bigger than infinity because you can just go on and on forever, you could compare an infinite plane (you just go on and on forever!) with an infinite volume, which presumably most people would call bigger.

Maybe most people would say that, but the two sets have the same cardinality (i.e. "size").

Sets?

[Charlie Reams]

Also I'm not sure if this is a sensible analogy or not, but against someone like me saying that you can't get bigger than infinity because you can just go on and on forever, you could compare an infinite plane (you just go on and on forever!) with an infinite volume, which presumably most people would call bigger.

Maybe most people would say that, but the two sets have the same cardinality (i.e. "size").

Sets?

The set of points that make up a plane, and the set of points that make up a volume. In fact the set of points on a line of length 1 has the same cardinality as a volume that is infinite in all three dimensions. Cardinality is counter-intuitive like that, but it's the only notion of size that works at all for infinity AFAIK.

[Gavin Chipper]

The set of points that make up a plane, and the set of points that make up a volume. In fact the set of points on a line of length 1 has the same cardinality as a volume that is infinite in all three dimensions. Cardinality is counter-intuitive like that, but it's the only notion of size that works at all for infinity AFAIK.

OK, I'm with you. What I meant was the size of the "space" rather than set of points if that makes any difference. However big a 2D area is, it's always smaller than a volume of any non-zero size, as it has no volume! So my anaolgy was that counting integers is like making a 2D area bigger and bigger. Yes, it's "unlimited" but you can still get bigger.