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

http://www.bbc.co.uk/h2g2/guide/A577875

This is an entry about the different sizes of infinities you get from set theory. Its quite a technical subject, but (with lots of help already from Arpeggio) I've tried very hard to make it accessible, defining all terms, lots of examples etc.

There is also an earlier version at: http://www.bbc.co.uk/h2g2/guide/A575598
...which is almost identical except I had smileys in as example objects in sets, which I'm guessing wouldn't make it into the guide.

I must confess I'm not good at maths, I mean I can count, a bit anyway. I always knew there was some point to maths other than being able to work out if I've been short-changed.......

Seriously, I like this, it's well written and it does all the hand-holding I need. I don't feel qualified to comment on it technically since reading it is the only time I've really looked at sets since school (many years ago), and back then I considered myself a success if I could remember which sign meant union and which meant intersection.

I haven't encountered many attempts to describe anything this complex in the guide so I say more power to your typing finger.

Cheers,

Cestus

I have to say that I have never seen Kantor's proof explained any better nor without a Hebrew character here or there.

Perhaps, you might give Georg Kantor's full name for those who want to look him up.

I saw one sentence in there that was missing a word, but I can't find it again, so it must have been these glasses. (t's very hard to search for missing words, even wild cards don't help much.)

I don't know that there is any way to simplify this anymore than you have though it might help if formulae were laid out in a table rather than in paragraphs. Still, once you get to the self-referencing bits all bets are off, any way.

It also might help to give some examples of infinite sets that are the same size even though they might seem like they should not be as well as infinite sets that are larger. (I mean as in a series of numbers as well as a formula that generates them.)

The important thing here, I think, is that there are classes or hierarchies of infinities. And I suppose that I might as well ask you to explain why we care. I can't explain, even do I do care and think the differences are 'neat.' Are there implications that make sense without math? Or, with just a bit of math?

I like the way this is written (didja really needta use 'kind'?) and I don't see how it could be explained any easier. Send it in for editing, already. (I'm dying to see the artwork for this one and it better not be two differently sized infinity signs. )

Barton

That should have said: didja really needta use 'kinda'? It really hurts me to type that word, I guess.

Barton

Added full name.

Infinite sets of same size which don't look like theyre the same size - I think that would go better in with Hilbert's hotel. There's a danger of covering too much and not leaving enough material for related topics to be done full justice without overlap.

Explaining why we care? Eeek. I think I'll just leave it as a 'wow' type thing. I don't know any real-world applications.

Changed 'kinda' to 'somewhat' - looks better that way.

I need more coffee before I can finish reading this!
Until then, Have I Got News For You - the missing words round...

"If we run of elements of X before we run out of elements of Y, then Y has more elements, pretty obviously" - should be "run OUT of elements of X".

Also, use of the word "onto" is (I think) not defined, and looks as though words have been missed out, e.g.
"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 sent to by f). "
Or did I miss something?

Other than the above, excellent. I love all this wacky maths stuff. Just need more coffee...

Fixed the missing 'out', and rearranged things a bit to make the definition of 'onto' clearer.

I know this is not in the same style as what you have written but I learned 'onto' in the sense of one set can have each of its elements mapped onto an element of the other set. The key word is 'mapped' in a similar sense to projection. Admitedly, it is applying a geometric (or geographic) image where there is no real cause, but I suspect that that is where the 'onto' originated.

Barton

Excellent entry. Congratulations.

Problem: Can everyone else see the '≤' properly??

When I looked into the entry 'Kantor' was spelled 'Cantor' but that is probably already corrected...

Cheers,

HELL

OOps...

Cantor IS actually the correct form... I just guessed, since the guy was German...

We apologise for the inconvenience,

HELL

I think the origin of the use 'onto' (for f from X to Y) might be something to do with the image of f being the whole of Y, so f goes 'on top of' Y rather than just into it. I dunno, very weird using the word so much then not being able to work out why it's called what its called.

Good article. There's one its which should be an it's near the end.

I found about 4 or 5, plus one 'its' that is an 'its' (possessive thingummy)

Fixed.

Super HenryS !

You really pulled it off. Unscary and not condescending is a trick, so consider yourself a magician.


Scout! Ohhh Scooo-uuut! [rummage in bag for Scout-Bait treats] would you rec' this already? This is the good stuff, made gooder so nonmathsy types can have fun with it. HenryS did a trrrriffic job and since I helped a bit with the midwifery, I want this on the Front Page where we can all kvel, now, already.


beauuuteeful baby you got there HenryS!

Arpeggio doing her best imitation of a N'Yawka, for no reason at all,
for LeKZ

Yes, it's a big, err... great entry!

It's been a few days since I had set theory but it's all back now

As to our usage (a 's 'we', as in 'how are we today?') of the word 'we', there's an entry at http://www.bbc.co.uk/h2g2/guide/A281521

(I know for sure there are more 'we's in conversations below that entry, but they are still hidden).


Bossel (Scout who has given up recommendable entries, the list grew to, err... big and is a database now)

Hmmm, surprised I haven't commented on this before.

Well, its excellent. Not much else to say really - should go in the guide.


Has anyone else noticed the weird bug in footnote 5? If you click on it and read it from the bottom of the page it reads correctly but if you just hold your mouse cursor over the superscript 5 in the main text (note this only seems to work on PCs in my experience) the 'less than or equal to' sign changes to just a '='.
Weird

Orcus

I guess that's the browser. IE and Opera seem to have a bug. Netscape hasn't because mouse hovering doesn't show anything!

Yes, Internet Explorere, at least, demotes those high ASCII number characters down. It also doesn't show link tagged words in the footnotes correctly. It's the browser's bug not anything on h2g2.

How about someone picking this thing already?

Barton

And an infinite number of them...
Your entry has been chosen for inclusion in the edited guide. No mean feat...
When the editing is complete, you should get an email, and everyone will see it on the front page!

J

* Waits for the infinitely prolonged party to begin *