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Aleph-0 bottles of beer on the wall, Aleph-0 bottles of beer,
Take one down, pass it around, Aleph-0 bottles of beer.
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Carpal Tunnel
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Your diet lacking in Omega-3s, Max?
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veteran
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old hand
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old hand
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Excellent. Wish I could get it to scan, though.
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Carpal Tunnel
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AL eph null BOT tls of BEER on the WALL
AL eph null BOT tls of BEER
What's not to scan?
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Carpal Tunnel
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What is that symbol. On my screen it looks like a severely wobbly N. Is it a combination of symbols?
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Carpal Tunnel
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OP
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It's Aleph, the first letter of the Hebrew alphabet.
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addict
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addict
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Shall cheerfully admit that this thread and tsuwm's link went clean over my head. Made me feel glad almost that I am mathematically challenged!
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Carpal Tunnel
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If infinity isn't over your head, your head is seriously in the clouds. And then some.
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old hand
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Hey maahey
Since neither my knowledge nor my explanatory powers are up to it, and I don't even know if you're interested, I will do as Anil Warner always recommended and 'bash on regardless'.
Alpeh-null, which is represented by the 'squiggly' figure in tsuwm's link, is the number representing what we, in layman's terms, call 'infinity'. This is the number of items in the set of number 1,2,3.... (continue counting for ever, adding one each time).
The mathematically interesting thing about this is that there are sets with more items in them than this 'normal' or 'everyday' infinity.
For instance, the set of rational numbers (all possible fractions made out of integers), has exactly the same number of members as the set of natural (or counting) numbers: Aleph-null. The set of irrational numbers, on the other hand (square root of two, three, five etc and e and pi) has 'infinitely' more members than Aleph-null. It is often (always? correct me some mathematicians) represented as Aleph-one.
All infinite sets, aleph-null upwards, have some peculiar properties, including this - aleph-null minus one is still equal to aleph-null. Hence the joke about the beer bottles.
Hope this helps.
cheer
the sunshine warrior
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Another thing to remember is that Georg Cantor, the mathematician noted for his work with infinities, spent his golden years in the state nuthatchery.
They say that his work didn't drive him there. They say…
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old hand
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they also say...
The set of all sets that do not include themselves
is a logical impossibility and hence demonstrates the impossibility of proving mathematics from first pricniples. At least, I believe this is what Russell and Whitehead came up against when they were writing their Principia, and hence had to give up. No wonder poor ol' Cantor went a little loopy.
cheer
the sunshine warrior
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If y'all are wondering about that set of all sets that do not include themselves, ask yourself. Does the set of all sets that do not include themselves include itself?
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old hand
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And, if the plurals sound confusing, try it as "the set of every set that does not include itself"...
etc
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Douglas Hofstadter spent seven hundred pages trying to make that clearer, and expanding on its implications. (After Godel did it first, more succinctly. But also CAUTION: NEOLOGISM ALERT! more jargonically.)
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addict
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Thank you Shanks! That *was helpful and I even indulged in a serious attempt to shake off the terrrible stigma that my mind has associated with numbers to ponder the points you and others made. drifting...Has anyone read RK Narayan's essay on higher mathematics from his collection of essays. It kept me in splits throughout; shall see if I can find a link.
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veteran
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Aleph-one. I had always seen it written as Aleph sub C. They knew it was bigger, but they didn't know if there were any infinities between Null and C, so they were reluctant to call it Aleph one.
I was discussing this with a professional mathematician colleague of mine and he said that they don't do this any more. He believes that someone has actually proven that C=1 (there are no infinities larger than aleph null and smaller than aleph C).
Caveat: He's a super smart guy, but I think his forte is probability theory and analysis of algorithms.
k
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I, too, had heard they weren't sure if the continuum was Aleph-1. Glad to hear they've got that straightened out. I was worried.
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old hand
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Maahey
R K Narayan evokes fond memories - Malgudi Days and so on. For some reason I have a vague notion that I also might have read the higher mathematics piece you speak of - but then again, maybe not. Would love to read it, if you can find a link.
cheer
the sunshine warrior
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I checked a few things on the web and found two accounts that disagree with each other and with my friend's interpretation. One says that it's been proven that C <> 1 and another that it's been proven that we CAN'T prove whether C = 1 or not. I'm a little busy right now and don't have the time to look into it, but I am intruiged. If I can get some time over the next few months, I'll check it out.
k
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That's an old book by George Gamow. He starts by telling a story of two mathematical naifs who can't count above three because they have no number for it, and then goes on to develop the theory of infinities:
--aleph-null is the number of integers/reals
--aleph-one is the number of rationals/points on a line or plane or solid or...
--aleph-two is the number of possible curves in space
--There is no larger infinity that we can conceive. C, the cardinality of the Continuum, may or may not be equal to aleph-two, or might be between them.
So at least where orders of infinity are concerned, we're no more advanced than his two naifs in the story.
The book is from the 'Forties, I think, and is a delightful read. Has the math changed since?
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journeyman
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Ooh, I know this one.
The relationship between aleph-1 and c is called is the Continuum Hypothesis. It's now known to be independent: you can choose it to be true or false, and either is consistent with standard mathematics.
The next infinity after aleph-0 is called aleph-1, then come aleph-2, aleph-3, ..., aleph-aleph-0, ...
Nothing is known that has size aleph-1 (or any of those others bigger than aleph-0).
The set of integers has the size aleph-0. The set of real numbers has a size called c (meaning continuum), bigger than aleph-0. The set of all subsets of the integers has a size that can be expressed as 2^aleph-0 (two to the power of aleph nought), which is also bigger than aleph-0.
So we have three infinities bigger than the smallest one: aleph-1, c, and 2^aleph-0. You can prove c = 2^aleph-0. Since aleph-1 is the next biggest, either c = aleph-1 or c > aleph-1. The Continuum Hypothesis, posed by Cantor, is that they're equal.
In 1938 Gödel proved it would be consistent if the CH was true. In 1963 Paul Cohen proved it would be consistent if the contrary was true (i.e. there were infinities strictly between aleph-0 and c).
P.S. Who needs infinities? They're mathematicians: when they get to nought bottles they keep going to -1. :-)
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Pooh-Bah
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As an aside, a recent article in The New Yorker reviewed a book concerning infinity and Cantor. It wasn't a very glowing review but not completely scathing either.
As for myself, infinity doesn't interest me much, but the highpoint of my month has been learning that i raised to the power of i (that is, i^i) not only has a value, but it is a real number and not a complex number. The wonderfully versatile Google can be used a calculator, and if you enter i^i and hit enter you get i^i = 0.207879576.
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(e to the i pi) minus one equals zero.
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Pooh-Bah
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Yes, that is one of the most beautiful and intriguing equations in all of mathematics.
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Or wait a minute. Is it (e to the i pi) plus one equals zero?
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Pooh-Bah
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Is that pumpkin or Dutch apple?
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