h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Picked this up from the peer review discussion on infinity:

"How many dimensions do I see in? The usual five, why? (It took me until the age of 38 to understand about the three everyone talks about... I assumed they meant five, and just called them three for convenience.)"

I'm really interested to try and understand what you mean by this. I've heard of people, with much practise, being able to 'visualise' 4 dimensional mathematical objects and sometimes above 4. Is this what you mean? Are you able to turn say, a hypercube (aka tesseract) around in your mind's eye? Or do you mean something else?

HenryS,

Thanks for the little line R proof. I thought so, and all those infuriating maths teachers just said 'No, you can't DO that'. They were trying to teach arithmetic, to children. I was... not a 'normal' child. What you explained was so intuitively obvious to me, that I should never have been able to explain it. This is partly because I didn't pursue higher maths (I should have done -- four separate physicists, on four unrelated occasions, told me I was a 'natural' at physics, and I *love* physics, but I just haven't the maths. My male parent, an extremely abusive person, happened to have doctorates in all the hard sciences -- yah, ALL -- and both my ex-brother and I had to get away from the hard sciences, or we'd have beeh forever and forever 'competing' with him. The ex-bro took up computers, which were new and a little intimidating to the male parent. I went into all the human sciences, and kept medicine as a sort of 'hobby'. This is all sort of relevant, I'm trying to work out how to make these perceptions make sense in a line of words...)

Okay. I have *never* encountered a mathematical concept I could not grasp intuitively. Not a physics one, either. Doing actual computation bored me comatose in school, and except for Algebra I and II, and Geometry, I had terrible marks. It was all so repetetive. The concepts just... make sense. The first time I bumped into *any* cognitive activity which gave me *any* trouble, it was Statistics for the Social Sciences, which I found very counter-intuitive. I was 22 the *first* time I did not understand a subject in school. I didn't KNOW how to deal with not understanding. Ultimately, I 'got' it. I just had missed the very obvious, but slightly unexpected point that all one can do with ordinary probability and statistics is 'fail to reject'. I was looking for Deeper Meanings that were not there. After that it settled down to a series of very tedious problem-sets.

'Tesseract'? As in Madeline L'Engle, right? A Wrinkle in Time was my all-time favourite book (of its kind -- I read voraciously, and in all areas) from the age of 6 until I was at least 10. I thought it made such perfect sense that I used to lie in bed, trying to align myself 'just right' with the Universe, so I could be Elsewhere. Whether that means I could 'see'-- wrong verb, it's more closely analogous to tactile or kinaesthetic sense -- yes, I could. I always have done. Mobius strips led me to immediately demand of the male parent (I think I was 4...hang on... not in school yet, some friend of the family had shown me the Mobius strip, so then I did it again, later, and then, commenting on the object existing in what looked like three dimensions, but having only one side, which meant it was 'really' two dimensions, I asked him, 'so how d'you square *this*?'

He was a brute, but he was also an educator. He asked me to try to imagine it, and I went on to give an amazingly good description, I found out when I was 19, of a Klein bottle, making it up as I visualised it. Now, I think I've squared that fairly comfortably, though it's not a 'shape' in quite the same way as a Klein-bottle could be. Natually, a Klein bottle squared would *seem* to exist in 5-space(time). It helped when I bumbled across the two separate definitions of Dimension 4, as either space-time, or temporal time, which I took to mean 'both' (like light being particle and wave, you use whichever angle of analysis is relevant or helpful; that does not change the nature of light). Klein-bottle squared didn't *work* so long as I was thinking of 4D as temporal time only. It works just fine if I thing of 4D as space-time.

I had a boyfriend, when I was 19, who was finishing up his undergrad work in physics and maths. He used to just shake his head and say 'how do you do it?' after I would propose an idea that was apparently Really Advanced. I dunno... I didn't really understand that other people did not do it. I've always perceived a lot more than the people around me, but it's only in the last year (I'll be 40 in Oct) that I have started to understand the *nature* of some of those perceptions, and why they've always seemed uncanny to people. When 'chaos theory' first showed up, I read everything, however technical and over my head (which it was, until Glieck's book) I could lay my hands on. THIS made the sense that conventional prob+stats did not. This is a realistic understanding of things, and one that, directly someone pointed it out, I could look at and say, Of course! So could my friends, so it never dawned on me that nonlinear equations could be a 'hard' concept.

In the last year, I have become much more conscious of the level of genii with whom I have interacted for most of my life. I always was 'different' in that singularly unpleasant way that 'average' kids have, for making gifted kids feel really crummy about themselves.

In 1994, I sustained a mild brain injury. It was mild, but every so often, I still bump up against cognitive tools I *used* to have, which are simply *gone*. Other children 'count sheep'. I did logarithms, changing the base for variety. Now, I'm not quite sure I could understand a logarithm if it were explained to me. I can't extract anything above square roots in my head any more, and not always those. Some ordinal functions went bye-bye, so I have trouble remembering which thing to multiply by which other thing, when doing metric conversions. In 7 years, I have adapted. In the last 7 years, I have 'looked things up' more than I did in my entire life until that time.

Humans have considerably more than five senses. The Others have to be analogised, because language does not accomodate them. The sense for perceiving electromagnetism is 'visual'. The sense for perceiving everything from radio to x-rays is 'auditory'. My colour-vision tests out freakish, in that infrared and ultraviolet are just names for colours. My auditory range is also extreme. And, as I said, the 'dimensional' sense is body-sense. The dimensions are there, and the body exists/inhabits them. 6D is very knotty and gives me a headache! At least, I *think* it's 6. It's the one I tried to get upsquares to from 5. That was intentional, and recent. The definition of 4 as space-time was also with that boyfiend. He was trying to explain the 'curvature of space', and one of those visible light-bulbs turned on over my head as that Klein bottle quietly squared itself. At that point, he decided to stop lecturing and take me out for a beer, because I had not attended college yet, and had dropped maths 2 years before finishing school, and he had summed up several of his most challenging classes from his senior year as an undergraduate, and not only was I interested, but also I could keep up, *conceptually*.

I do not pretend I could do *any* of the equations, and I know I should not enjoy trying. I know part of it is an emotional block, because the male parent was actually quite a Famous Person in the sciences, and was a vicious, filthy sociopathic piece of scum. Part of it now, is I have trouble with the *names* for ==>this way, and <== that way. I have no trouble telling them apart, but the names went with some of the perfect recall, a lot of the maths, and some ordinatory function. My absolute pitch is unreliable, too, now. It is there most of the time, and sometimes, it's like that pen, that I *just* had... O, and absolute pitch is *definitely* kinaesthetic, not auditory. I know how the sound resonates in/with the body. That is not a function of 'hearing'.

I've never tried to say any of this in writing before. In fact, I'm rather surprised I posted that remark. That sort of thing tends to put people off in the worst way. I was soo overtired last night... so my judgment was off.

If you have other questions or want to tell me cool things about infinity, I really love infinity! I might have been big enough to sit up by myself the first time I heard of it. It's been a friend ever since!

Thanks for the interest,

Arpeggio, for LeKZ

Arpeggio:
"all those infuriating maths teachers just said 'No, you can't DO that'."
I remember that one when it was in primary school and the teacher was telling someone that you can't take the larger number away from the smaller number. Lots of maths is looking at something you thought was impossible, and working out how to think about things so it is possible
"Okay. I have *never* encountered a mathematical concept I could not grasp intuitively."
You're lucky there. It takes a little while for me with new things, and some things there's never an intuitive way to think about them, but its certainly the case that I enjoy and do best at the ones that do come intuitively. And the ones that come most intuitively are the visual ones for me - so I'm going into topology which has lots of pictures

"'Tesseract'? As in Madeline L'Engle, right?"
Not sure, I've not read it. Tesseract as in: http://www.bbc.co.uk/h2g2/guide/A510986
(I did the picture at the bottom)
"Whether that means I could 'see'-- wrong verb, it's more closely analogous to tactile or kinaesthetic sense -- yes, I could."
This is probably going to be impossible to describe to me. I was talking to someone with synaesthesia a week or so ago - bet its the same sort of thing: if you don't have it theres no way to imagine what its like, and if you do, theres no way to imagine what its like without.
(Moebius strip)"'so how d'you square *this*?'"
Theres a number of things I imagine you might mean by that...by analogy with going from a line to a square or from a square to a cube (increase dimension of the thing (manifold) in some way), or in the sense of 'closing off the edges', like going from a cylindrical strip of paper to a torus by joining the edges together?

Ah, Klein bottle...then you mean as in joining the edge to itself and 'closing it up'.
"two separate definitions of Dimension 4, as either space-time, or temporal time"
Then theres the third definition of a space with four dimensions, the mathematical definition, which just assigns four coordinates to a point rather than 3. So a point is specified as (x,y,z,w).

"I always was 'different' in that singularly unpleasant way that 'average' kids have, for making gifted kids feel really crummy about themselves."
I've never been one for peer pressure, and where I am now (Oxford uni) theres enough clever/weird people about that its almost the norm

"The sense for perceiving electromagnetism is 'visual'. The sense for perceiving everything from radio to x-rays is 'auditory'."
Not sure quite what you mean there. I don't think humans have any way of directly perceiving radio waves.
"And, as I said, the 'dimensional' sense is body-sense. The dimensions are there, and the body exists/inhabits them. 6D is very knotty and gives me a headache! At least, I *think* it's 6. It's the one I tried to get upsquares to from 5."
to me (not really visualising any of it much) theres not much difference between 6 and any higher number of dimensions. The maths gives you ways to think about and work with the objects without having to 'see' what they look like. And given a lot of time and practise you can get a better feel for how things work. A lot of it is still by analogy with 2D and 3D things we can visualise though.
Incidentally, I've heard it said that much of the really interesting stuff goes on in 3D and 4D. With 5 and above theres 'too much space' and its too easy to move things about. Since hearing this I've always suspected that there is interesting stuff going on up there, but we don't have the intuition to know what the interesting things worth looking at are. Oh well.

"If you have other questions or want to tell me cool things about infinity, I really love infinity! I might have been big enough to sit up by myself the first time I heard of it. It's been a friend ever since!"
Have you heard about the different sizes of infinity? Now they are cool - set theory type stuff. I probably ought to get around to writing an entry about them...

HenryS-

I'm going into topology which has lots of pictures -- If you're essentially a visual thinker, and the majority of humans are, that would be really interesting! It sounds like fun. I, as usual, have a pretty solid grasp of the *concept*, and not even information on the How They Do That part. I'm guessing, but it seems to me like a 3 or 4D (thanks for 4-space, that was new ) application of the Calculus? I never actually studied the Calculus, but aforementioned boyfriend when I was 19 explained it. After he had explained for a few minutes, I say 'Oh! It's just needlepoint!'. You are probably making precisely the face he did.

Needlepoint is done on canvases with different 'gauges' -7 holes per inch to 14 holes per inch. A curvilinear design on low gauge canvas is much less accurate than if it is done on high gauge canvas. I think the rest is self-explanatory.

I can visualise the same conceptual structure upsquares in 3, or even 4D without too much trouble.

Are you studying topology as applied maths, or as... brain( )gap... the other word... strictly theoretical and trying to improve upon the existing body of knowledge?

Have you heard about the different sizes of infinity? Now they are cool - set theory type stuff. I probably ought to get around to writing an entry about them... Nope, I haven't. Practise writing that entry by telling me ! These are the things that keep life fun and happy: learning cool theories about the 'sizes' of infinity. The spiritual side of me says, 'Yeah, right, the mathmos have figured out God's shoe size...I'm sooo sure.' If God wants humans to know God's shoe size, then they can do it. Why not?

(Moebius strip)"'so how d'you square *this*?'"by analogy with going from a line to a square or from a square to a cube (increase dimension of the thing (manifold) in some way), or in the sense of 'closing off the edges', like going from a cylindrical strip of paper to a torus by joining the edges together?

Ah, Klein bottle...then you mean as in joining the edge to itself and 'closing it up'.
What I was thinking when I asked, was going from a square plane to a cube. It wouldn't, in my head. Well, yes, no, and both. The 'torus' idea never really occurred to me. I was just a little kid and making it up as I went along, and what I got was a shape where the inside and the outside were the same side, and enclose each *other*. I later described this and Frank (the mathsy boyfriend at Yale) called it 'Klein bottle'. Now I'm not so sure, but the things are great fun, whatever they're called. Now I'm mentally doing unspeakable things to doughnuts but all I get is snake-swallows-own-tail-and-turns-self-inside-out cyclical repetitions.

I've never been one for peer pressure,
No. Peer pressure bites. But Oxford is terrific. What part of your degree programme are you doing at the moment?

Not sure quite what you mean there. I don't think humans have any way of directly perceiving radio waves.
Neither do I. Trouble is, during my teens and twenties, I could start singing something, and then turn on MTV or the radio, and it would be at the same point in the same song. If this had happened 50 times, I would still call it coincidence. It happened *all* the time, to the point where it brought out the superstitious side of the female parent. It became an ongoing party-trick, which my friends loved, and which frankly scared me: someone would fiddle with the radio tuner, and change the frequency with it OFF. Then, they'd wait, and I'd wait, until I started singing, and a few lines into the song, on would be the radio. The number I missed is so statistically improbable that there *must* be an intervening variable we always overlooked. I almost *never* missed, and when I did, the radio was inevitably staticky. I have never been able to do this with AM radio stations. Local tv, yes, and FM, yes. Since they changed to stations which used formats of music which I didn't always like (disco, pop rock) and never listened to on purpose, *I* have no idea how or why this worked. Theories?
O... I can still do it some, but now I think it falls inside about two standard deviations from coincidence, and I am willing to accept coincidences that are wildly improbable, over anything that is downright inexplicable.
Yes, I had to 'listen' for the music, but to me, it never sounded as though the radio was OFF. No, I am not subject to auditory hallucinations as specific as what is *actually* playing on the radio -- or indeed specific at all. I get low-level auditory hallucinations if I am very overtired, but fatigue-poisons explains them adequately, specially because I am an auditory thinker, for the most part.

With 5 and above theres 'too much space' and its too easy to move things about. Since hearing this I've always suspected that there is interesting stuff going on up there, but we don't have the intuition to know what the interesting things worth looking at are.
Um, I do, to some extent. 'Too much space' is only if one doesn't see/feel/be in the next ones upsquares. I wouldn't know. I expect I should find 3D positively claustrophobic by the narrowness of its definitions. *Why* I do have this intuition is a looong story, and very complicated. But either I really do, or I am completely convinced of something that is utterly nuts. I am basically a logical positivist, so I try not to fantasise 'uncanny' in, where hyperalert/hyperaware might do the job. There have been theories. Since I don't know how *not* to see this way, I always assumed everyone did. It was last year that I realised, This is Not Normal. They mean *just* THREE. I mean three and all the *other* stuff. Like in-between dimensions: 3.67 is also real. It's all Very Big, and utterly impossible to articulate meaningfully, so I usually don't try.

Let's hear some of those infinite 'sets'!

Arpeggio, for LeKZ

Topology - visual thinkers: I'm more visual than most, at least looking at the other mathematicians here. I can do things like logic and algebra, but its the visual type things which are the easiest to remember/work out proofs for.

Calculus as done at university pretty quickly gets defined for arbitrary numbers of dimensions (even infinite numbers of dimensions), but thats not really topology, though its certainly related. Topology is ...umm...difficult to explain. Well, lemme approach it like this: (stop me if you've heard this before. oh except forum posts don't work like that ) The standard joke goes that a topologist can't tell the difference between a doughnut and a coffee cup. This is because they've both got one hole through them, and you can continuously deform (i.e. squidge) one to look like the other. The idea behind lots of maths is to try and make it easier to look at something by ignoring details irrelevant to what you're interested in - topology ignores lengths and angles, and is only really interested in things like connectedness and continuous stuff.

Needlepoint - I sortof feel like I know what you're getting at but I'm not sure. When he was talking about it did he mention things like 'epsilons' and 'deltas'?

BTW, I think I like the word 'upsquares', but I'm not entirely sure what it means. Any pointers? Its up a dimension? But the analogous concept in the next dimension, in some sense?

Pure maths, not applied maths. Yes, to increase the body of knowledge rather than to apply it.

Right. Infinities. I'll assume you believe in the existence of N, the set of all natural numbers, {0,1,2,3,....}. So how many elements does N have? Clearly its not any finite number of elements, so it must be (by definition) infinite. Lets write |X| for the 'size' of any set X. If X is finite, with (say) 42 elements, we would say |X| = 42, for infinite sets we haven't yet given these 'sizes' (jargon: 'cardinalities') names, so lets just write |N| for the size of N. Now the question is, are there any sets X which have 'more' elements than N? I.e. is there some X such that |X| > |N|?

Well first we need to say what we really mean by 'more'. We need to do this in a way such that we don't need to count the elements of the sets (which is kinda difficult for infinite things). To say that X has less than or an equal size to Y (i.e. |X| <= |Y| ), we can try pairing up each element of X with an element of Y. If we run of elements of X before we run out of elements of Y, then Y has more elements, pretty obviously. To formalise this, we need a function f taking elements of X to elements of Y, which is 'one-to-one', meaning that for a and b different elements of X, they will map to different elements of Y (f(a) is different from f(b) ). Here we are pairing up each a in X with its f(a) in Y, and if we can do that for all the a in X, then we must run out of elements of X before we run out of elements of Y.

So that's what 'more' means, defined in a way that doesn't require we count the things. We can say
|X| = |Y| if we have both |X| <= |Y| and |Y| <= |X|. It is possible to prove that with one-to-one functions f from X to Y and g from Y to X you can make a one-to-one and onto function h from X to Y (onto means everything in Y gets mapped to by something in X), which means that h pairs up each element of X with exactly one element of Y, and everything gets paired with something else, so X and Y really must be the same size. This proof however (the Schroder Bernstein theorem), is beyond the scope of this article.

All this is pretty obvious for finite sets. What is maybe less obvious is that for infinite sets, you can have such a function f (from N to N say) which is one-to-one, but not onto (there can be some things that nothing gets mapped to). For example, if the function f just doubles a number, then (pretty obviously) if a is different from b, then 2*a is different from 2*b, so f(a) is different from f(b). I.e. f is one-to-one. But (for example) nothing maps to 3. This sort of thing cannot happen with finite sets. For more fun along these lines, look up Hilbert's Hotel at http://www.bbc.co.uk/h2g2/guide/A414523

Anyway, back to the plot: next we define a way to get a new set from one we are given, and show that the new set is really bigger than the old one:

For a set X, the power set of X (written P(X) ) is defined to be the set of all subsets of X. So is X were the two element set {a,b}, then P(X) is the set containing all the subsets. I.e.
P(X) = { {a,b}, {a}, {b}, {} }
Note that last element, the empty set, *is* a subset of X. Its the subset not containing any elements at all.

For finite sets, its not too hard to show that |P(X)| = 2^|X| and so |X| < |P(X)| (that's a strict less than - in terms of our functions definition, |X| < |Y| means that there is no possible one-to-one function from Y to X. I.e. Y is just too big to 'fit inside' X. If you think about what this says for finite sets, you'll see that this makes sense as a definition. Ok here's the hard bit: we can show that |X| < |P(X)| for *all* sets - not just finite ones. Here's the proof, first worked out by Cantor:

First we can pretty easily show that |X| <= |P(X)| by taking the function F(x) = {x} (F takes each x in X to the subset of X consisting only of x). This is obviously one-to-one.

We now want to show that there is no possible one-to-one function from P(X) to X. Well suppose we did have such a function f and we'll get a contradiction and show that this f is impossible. Here's the clever self-referential bit:

Define a subset S of X by:
S = {a in X such that there is some subset A of X with f(A) = a, and a is not in A}

In other words, S consists of all elements of X that something gets mapped to by f and that the thing they get mapped to doesn't contain the thing you map.

Now S is a subset of X, so it is a member of P(X), so we can see where it gets mapped to under our function f. Let s = f(S). Now the question is wether or not s is in S.

Suppose s is in S. Then by the definition of S, there is some subset A of X with f(A) = s, and s is not in A. Well we know what that A is. We know that f is one-to-one so f(A) = s = f(S) so A = S, and s is not in A, so s is not in S. But we started this paragraph assuming that s *is* in S. Contradiction, we can't have s is in S.

Now assume s is not in S. Well there is some subset A of X with f(A) = s (we can take A = S), so for s to not be in S, the last bit "a is not in A" must be false. So "s is not in A" is false. So "s is not in S" is false. So s *is* in S. Another contradiction. So the whole thing doesn't work - we must be unable to find that function f in the first place.

That's Cantor's proof that |X| < |P(X)|. So if we stick N (the natural numbers) in there, we get |N| < |P(N)|. N is an infinite set, so P(N) must be infinite as well, but its also strictly bigger than N. So it must be a bigger infinity. It turns out that |P(N)| = |R|, the number of points on the real number line. Are there any bigger infinities or have we hit the top yet? Well, we can use Cantor's proof again to show that
|P(P(N))| > |P(N)|, and again to show |P(P(P(N)))| > |P(P(N))|. And so on, to get infinitely many bigger and bigger infinities. Which 'infinitely many' I hear you ask? Cantor's proof will get you a different one for each natural number, just by putting that many P's before N. but there are even more than that. There are unimaginably many different sized infinities. For any infinite number you can find more than that many different infinities. It's even provably impossible to form the set of all infinite numbers (in order to try and find out how big it is), its just 'too big' to even be a set, the set axioms break down with collections of things this big. Its big.

----------------
Does that make sense? I'll go remove a couple of those 'I's and stick it in an entry...

Klein bottle: You might be thinking of something like that, but its difficult to know. If you look at a small bit of it (ignoring the large scale structure) does it look like a bit of the plane or a bit of 3D space?
Oxford part of degree program: The part at the end after finals have finished and I'm lazing around in the not-sun (grr) punting, messing about in the park, reading and spending ages on h2g2 I'm off to Stanford in September to start my maths PhD.

Radio weirdness: might vaguely be related, I heard of someone with metal fillings in their teeth, who picked up radio stations on them. Not sure if it was FM or AM, and it wouldn't explain the fiddling with the tuner.

3.67 dimensions... there is a bit of maths that talks about non-integer dimensions - fractals. Dunno if thats anything like what you're talking about though. Idea here is that if you've got a 2D square, then if you scale it up by a linear factor of 2, you need 4 copies of the square to make the scaled up version. And with a 3D cube, you need 8 copies. So we can say the dimension of the object is log to base 2 of the number of copies you need of the object to make the 'linear scaled up by 2' version. This gets fractional when you get something like the Sierpinsky triangle (a web serach will find a picture of it) which needs 3 copies of it to make a scaled by 2 version, so it has fractal dimension of about 1.58.

(yay, another enormous post - made it into the longest posts bit of /info for the first time yesterday )

'Bigger and Bigger Infinities' article at http://www.bbc.co.uk/h2g2/guide/A575598
Suggestions please

Hullo HenryS,

I looked at the actual entry, and it's almost the same as this, so I'll just introject a few comments in ALL CAPS (sorry, it's a formatting problem...). I understood all of it just fine, and I am not keen on anything that is almost all equations. If you want this to be non-mathmo-friendly, you're going to have to add substantial bits in good ole English.
__________________
I THINK YOU NEED A LITTLE INTRO ON WHAT THE ARTICLE IS ABOUT, RATHER THAN STARTING BANG INTO THE ACTUAL PROCESS OF THE PROOFS.

I'll assume you believe in the existence of N, the set of all natural numbers, {0,1,2,3,....}. So how many elements does N have? Clearly its not any finite number of elements, so it must be (by definition) infinite. Lets write |X| for the 'size' of any set X. If X is finite, with (say) 42 elements, we would say |X| = 42, for infinite sets we haven't yet given these 'sizes' (jargon: 'cardinalities') names, so lets just write |N| for the size of N. Now the question is, are there any sets X which have 'more' elements than N? I.e. is there some X such that |X| > |N|?
OBVIOUS TO MATHMOS. NOT TO OTHERS. YOU'D NEED TO EXPAND. [EG. SYMBOLS FOR GREATER-THAN-OR-EQUAL-TO ETC. ARE NOT THINGS EVERYONE KNOWS HOW TO READ.]
THERE ARE TRIVIAL PUNC. AND SPELLING ERRORS IN THE ACTUAL ENTRY. WILL BE HAPPY TO SHOW WHERE TO FIX, IF YOU WANT.

Well first we need to say what we really mean by 'more'. We need to do this in a way such that we don't need to count the elements of the sets (which is kinda difficult for infinite things). To say that X has less than or an equal size to Y (i.e. |X| <= |Y| ), we can try pairing up each element of X with an element of Y. If we run of elements of X before we run out of elements of Y, then Y has more elements, pretty obviously. To formalise this, we need a function f taking elements of X to elements of Y, which is 'one-to-one', meaning that for a and b different elements of X, they will map THIS BIT IS NOT VERY CLEAR, USE OF 'MAP' MATHS JARGON? to different elements of Y (f(a) is different from f(b) ). Here we are pairing up each a in X with its f(a) in Y, and if we can do that for all the a in X, then we must run out of elements of X before we run out of elements of Y.
LAST SENTENCE EXPLAINS, SO PUT IT IN WHERE THE LAST CAPS ARE?

So that's what 'more' means, defined in a way that doesn't require we count the things. We can say
|X| = |Y| if we have both |X| <= |Y| and |Y| <= |X|. It is possible to prove that with one-to-one functions f from X to Y and g from Y to X you can make a one-to-one and onto function h from X to Y (onto means everything in Y gets mapped to by something in X), which means that h pairs up each element of X with exactly one element of Y, and everything gets paired with something else, so X and Y really must be the same size. This proof however (the Schroder Bernstein theorem), is beyond the scope of this article.
F,G, AND H THERE WERE A LITTLE CONFUSING...

All this is pretty obvious for finite sets. FOR YOU, YES. What is maybe less obvious is that for infinite sets, you can have such a function f (from N to N say) which is one-to-one, but not onto (there can be some things that nothing gets mapped to)WHY?. For example, if the function f just doubles a number, then (pretty obviously) if a is different from b, then 2*a is different from 2*b, so f(a) is different from f(b). I.e. f is one-to-one. But (for example) nothing maps to 3. This sort of thing cannot happen with finite sets. For more fun along these lines, look up Hilbert's Hotel at http://www.bbc.co.uk/h2g2/guide/A414523
HOW DID '3' GET INVOLVED? WE WERE TALKING ABOUT FS OF A, AND FS OF B, RIGHT? WHAT HAS THAT GOT TO DO WITH '3', OR IS THAT WHY NOTHING 'MAPS TO' IT? EXPLAIN MORE ABOUT WHY IT CANNOT HAPPEN W/ FINITE SETS, FOR READERS WHOSE INTUITION IS LESS GOOD THAN MINE. I *THINK* I GET IT, BUT THAT '3' IS A PROBLEM.

Anyway, back to the plot: next we define a way to get a new set from one we are given, and show that the new set is really bigger than the old one:

For a set X, the power set of X (written P(X) ) is defined to be the set of all subsets of X. So is X were the two element set {a,b}, then P(X) is the set containing all the subsets. I.e.
P(X) = { {a,b}, {a}, {b}, {} }
Note that last element, the empty set, *is* a subset of X. Its the subset not containing any elements at all.
YAH. I REMEMBER THIS BIT FROM SCHOOL. DOESN'T P(X) = { {a,b}, {a}, {b}, {} } PROVE P(X)>(X) BY ITSELF? I MEAN, IF X={a,b}, THAT IS ONLY ONE BIT OF P(X), KIND OF OBVIOUSLY?

For finite sets, its not too hard to show that |P(X)| = 2^|X| and so IF THE PROOF ISN'T HARD, SHOW? I'D WANT TO SEE WHY, RATHER THAN TAKE YOUR WORD |X| < |P(X)| (that's a strict less than - in terms of our functions definition, |X| < |Y| means that there is no possible one-to-one function from Y to X. I.e. Y is just too big to 'fit inside' X. If you think about what this says for finite sets, you'll see that this makes sense as a definition. EXCELLENT. VERY CLEAR. Ok here's the hard bit: we can show that |X| < |P(X)| for *all* sets - not just finite ones. Here's the proof, first worked out by Cantor:

First we can pretty easily show that |X| <= |P(X)| by taking the function F(x) = {x} (F takes each x in X to the subset of X consisting only of x). This is obviously one-to-one. IT IS?
'TAKES...TO SUBSET...x)' IS JARGON AGAIN. I HAD TO LOGIC IT OUT.

We now want to show that there is no possible one-to-one function from P(X) to X. Well suppose we did have such a function f and we'll get a contradiction and show that this f is impossible. Here's the clever self-referential bit:
IT IS VERY CLEVER, BUT THIS BIT HAS TO BE SPELT OUT, AS IF TO A SLOWER THAN AVERAGE 8 YEAR OLD, OR IT WILL ZING RIGHT OVER EVERYONE'S HEAD.

Define a subset S of X by:
S = {a in X such that there is some subset A of X with f(A) = a, and a is not in A}
EXPLAIN WITH NUTS, OR SOMETHINGS NON ALPHANUMERIC, THAT PEOPLE CAN 'SEE', OR THAT BIT IS REALLY HARD. ACTUALLY USING THINGS IS PROBABLY A GOOD IDEA THROUGHOUT, AS ENGLISH EQUIVALENT STATEMENTS TO THE EQUATIONS.
In other words, S consists of all elements of X that something gets mapped to by f and that the thing they get mapped to doesn't contain the thing you map. NO HELP... AND DOUBLE NEGATIVE

Now S is a subset of X, so it is a member of P(X)REMIND WHY, so we can see where it gets mapped to under our function f. Let s = f(S). Now the question is wether or not s is in S.WHY IS THAT THE QUESTION?
I'M BEING THICK ON PURPOSE, OBVIOUSLY. YOU SHOULD SEE WHAT I SAID ABOUT THE COMPUTER-GEEK JARGON! 'UNIX has a desktop? What for?'

Suppose s is in S. Then by the definition of S, there is some subset A of X with f(A) = s, and s is not in A. NOT CLEAR. Well we know what that A is. NO, I FORGOT. We know that f is one-to-one so f(A) = s = f(S) so A = S, and s is not in A, so s is not in S. But we started this paragraph assuming that s *is* in S. Contradiction, we can't have s is in S. USE EXAMPLES. YES, RODENTS ARE FINITE, BUT IF YOU USE SUBSETS OF RODENTS TO SHOW HOW THE CONCEPT WORKS, THESE EQUATIONS MIGHT NOT BE SO TRICKSY. I'M NOT VISUAL, AND I HAD TO DRAW THIS BIT, TO WORK IT OUT.

Now assume s is not in S. Well there is some subset A of X with f(A) = s (we can take A = S), so for s to not be in S, the last bit "a is not in A" must be false. So "s is not in A" is false. So "s is not in S" is false. So s *is* in S. Another contradiction. So the whole thing doesn't work - we must be unable to find that function f in the first place. TRY PHRASING THAT WITH FEWER NEGATIONS? 'WE MUST BE UNABLE TO FIND...' IS FUNKY. AND I JUST REALISED RODENTS ARE FINE, BECAUSE THIS IS X, NOT N, SO PLEASE RODENTISE, OR VEGIFY, OR OTHERWISE HELP OUT THE NONMATHSY READER.

That's Cantor's proof that |X| < |P(X)|. So if we stick N (the natural numbers) in there, we get |N| < |P(N)|. N is an infinite set, so P(N) must be infinite as well, but its also strictly bigger than N. So it must be a bigger infinity. It turns out that |P(N)| = |R|, the number of points on the real number line. Are there any bigger infinities or have we hit the top yet? Well, we can use Cantor's proof again to show that
|P(P(N))| > |P(N)|, and again to show |P(P(P(N)))| > |P(P(N))|. And so on, to get infinitely many bigger and bigger infinities. Which 'infinitely many' I hear you ask? NOPE, I SAID 'HOW MANY INFINITELY MANY?' NOT 'WHICH'... Cantor's proof will get you a different one for each natural number, just by putting that many P's before N. but there are even more than that. AND THAT LAST BIT IS PLEASANTLY OBVIOUS TO EVERYONE, BECAUSE OF HOW WELL YOU SET IT UP. There are unimaginably many different sized infinities. For any infinite number you can find more than that many different infinities. It's even provably impossible to form the set of all infinite numbers (in order to try and find out how big it is), its just 'too big' to even be a set, the set axioms break down with collections of things this big. Its big.
PROBABLY BEST TO EITHER EXPLAIN SET AXIOMS, AND WHY THEY BREAK DOWN AT THE ULTRAHYPERMEGALEVEL, OR NOT MENTION THEM AT ALL. I FIND THE WAY MATHS BREAKS DOWN AT THE QUANTUM LEVEL, AND AT THE OUTER REACHES OF BIG VERY INTERESTING, BUT THAT'S A PAPER ON ITS OWN.
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Does that make sense? I'll go remove a couple of those 'I's and stick it in an entry...

__________________
Actually, given that you did not provide any substantive 'fingers' to count on, it makes amazingly good sense. Your flow from the specific to the abstract and the finite to the infinite is excellent. The organisation and presentation of the steps is perfect. It's just the absence of mental 'fingers and toes' which makes this a Scary Thing for non-mathmos. You are so used to 'seeing' a and fs of b, and Ps of other things, that other people do not call language, that you don't see where it is not obvious.

Suppose you looked at something written in English, clearly, mais toutes les examples sont en Français clair, that's what you have here. Anyone who can read both does not even notice the transition, mais si tu ne lis pas Français, tu n'as pas d'idée what I just said. I could read something which switched languages like that, and probably not notice the transitions. I grew up switching between English, Hindi, and French in mid-thought.

That's the whole story on that one. I love it, and I'd like to see the rest of the non-mathsy world get a chance to understand it!

Really, excellent job.

I'll reply to the other part of your note tomorrow, ok?

TTYL,

Arpeggio, for LeKZ

Added quite a lot to the article, including some 'fingers'. Not quite sure they work that well. Let me know what you think.

Hulooo HenryS,

What happened to you? Your article should be ready for Peer Review by now. I keep looking for it.

Is is just end-of-term-wrap-up, or something?

Wondering if it's the smell in here or what?


Arpeggio, for LeKZ

Its been knocking around in PR already at: http://www.bbc.co.uk/h2g2/guide/F48874?thread=121896 I guess its a good sign for PR if there's so much going on that you can miss an entire thread :) Seems to have been received well and will presumably get picked up by someone sometime.