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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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Carpal Tunnel
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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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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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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
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