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oroboros commented on the word �? alef
In mathematics alef designates a particular kind of infinite set.
February 22, 2007
sionnach commented on the word �? alef
Didn't Cantor end his days in a lunatic asylum? Maybe peering into the abyss of the infinite got to him in the end.
A slight nitpicky comment. My very dim recollection of such matters is that it is possible to show that there are "more" real numbers than there are integers (in the sense of not being able to establish a 1-to-1 correspondence), although both sets are infinite. Thus, the cardinality of the real numbers is greater than that of the integers. I am used to seeing the first denoted by alef_1 and the second as alef_0 (where the '1' and '0' are subscripts). Then, Cantor showed that alef_1 is actually equal to 2 raised to the power of alef_0.
way more than anyone wanted to know - sorry!
February 22, 2007
oroboros commented on the word �? alef
Good recall sionnach! If you're interested here's a book to lay your hands on: "Elements of the Mind, the inner life of music and mathematics" by Rothstein, wherein he touches on the alef sets and also the Dedekind Cut et al. Especially good if you're also into music.
February 22, 2007
seanahan commented on the word �? alef
I believe the correct spelling is aleph, at least in English.
An even more nitpicky comment, Cantor never showed that aleph_1 is 2^(aleph_0). This is called the continuum hypothesis. Cantor proved that 2^(X) > X, and aleph_1 > aleph_0.
February 23, 2007
sionnach commented on the word �? alef
You mean aleph_1 might not equal 2^(aleph_0)?! I feel the whole basis of my Weltanschauung crumbling around me. Noooooo.....
Next thing you'll be trying to convince me that Zorn's lemma is not equivalent to the axiom of choice. (hopelessly abstruse joke)
February 23, 2007
seanahan commented on the word �? alef
That's some quality sarcasm right there. Even though I didn't remember what Zorn's lemma was, I figured out from context what you meant.
February 23, 2007
oroboros commented on the word �? alef
Did you know there's a rock group named "Axiom of Choice"?
February 23, 2007
sionnach commented on the word �? alef
Well, I'm nothing if not stubborn. What I stated yesterday was correct; the confusion may be one of notation. Getting away from the alephs for a while, it is a true statement that the cardinality of the real numbers is 2 raised to the power of the cardinality of the integers. The confusion arose (I suspect) because I used aleph_1 to denote the cardinality of the reals - this is not universally accepted notation, because some authors use aleph_1 to denote the next 'order of infinity' above aleph_0. It is not known whether in fact that this next order of infinity does equal the cardinality of the reals - the supposition that it is is the continuum hypothesis.
Summing up, if aleph_0 is the cardinality of the integers, aleph_1 is the next order of cardinality, aleph_R is the cardinality of the reals, it is known that aleph_1 < or = aleph_R = 2^(aleph_0). The continuum hypothesis is that that first inequality is actually an equality.
thank God; I was headed straight into some kind of existential funk.
February 23, 2007
oroboros commented on the word �? alef
Bravo!
February 24, 2007