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The Unicode chart for "letterlike symbols" states that

א 2135 ALEF SYMBOL = first transfinite cardinal (countable)

ב 2136 BET SYMBOL = second transfinite cardinal (the continuum)

I thought that the first transfinite cardinal is $\aleph_0$, the second is $\aleph_1$, and the continuum hypothesis is undecidable.

Which mathematicians did the Unicode Consortium consult when they decided to include these characters? Was this the standard usage when the first version of Unicode was developed in late '80s?

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The Unicode names for many symbols are erroneous, and unfortunately cannot be changed. Don't place too much value on them. –  Zhen Lin Sep 6 '11 at 12:49
    
@SN That's discussed in detail here math.stackexchange.com/questions/9475 –  Yuji Sep 6 '11 at 12:55
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The text you quoted is incorrect, but it might be relevant to mention that the symbol $\beth_0$ is used for $\aleph_0$ and the symbol $\beth_1$ is used for $2^{\aleph_0}=\mathfrak c$. –  Samuel Sep 6 '11 at 12:58
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For what it's worth, there's also a Unicode character whose name is (and will always remain) "PRESENTATION FORM FOR VERTICAL RIGHT WHITE LENTICULAR BRAKCET". –  Tanner Swett Sep 6 '11 at 20:30

1 Answer 1

up vote 4 down vote accepted

The use of $\aleph$ is quite arbitrary in that context, especially outside set theory.

I have seen $\aleph$ used as a variable for a general (well orderable) cardinal, as a symbol for the continuum, and it does not surprise me that someone whose mathematical background may not include set theory would confuse it for the first transfinite cardinal (i.e. $\aleph_0$).

However indexed $\aleph$ symbols are of course quite well understood as $\aleph_\alpha$ being the $\alpha$-th cardinal.

The $\beth$ symbol is used in mathematics a lot less often, it is the cardinality of the power sets, that is:

  • $\beth_0 = \aleph_0$;
  • $\beth_{\alpha+1} = 2^{\beth_\alpha}$;
  • If $\lambda$ is a limit ordinal, then $\beth_\lambda = \sup\{\beth_\alpha\mid\alpha<\lambda\}$.

I have not seen these used quite often outside set theory, and not often within set theory. As with $\aleph$ symbols, indexed notation has a clear definition.

$\beth_1=|\mathbb R|$, so the continuum hypothesis is $\aleph_1=\beth_1$. If you think on $\aleph$ and $\beth$ as class functions from the ordinals into the cardinals, then the Generalized Continuum Hypothesis asserts that $\aleph=\beth$.

Regardless to all that, coming from the Hebrew side of the screen, $\aleph$ is the first letter of the Hebrew alphabet, and $\beth$ is the second. I do not recall the Hebrew alphabet being ever referred to as "transfinite cardinals" though :-)

(Another interesting addition is that there exists a $\gimel$ (Gimel) function as well in cardinal arithmetics, $\gimel(\kappa)=\kappa^{\operatorname{cf}(\kappa)}$. Gimel is the third letter of the Hebrew alphabet following $\aleph,\beth,\gimel$. Seems that cardinal arithmetic is a lesson in Hebrew, much like mathematics is a lesson in Greek)

(One should be aware that it can be an apparently quite the confusion about the continuum hypothesis, and some mathematicians believe that $\aleph_1$ is defined as $2^{\aleph_0}$)

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Thanks Asaf; of course the Unicode has the whole Hebrew letters separately. The aleph and bet are doubly registered in Unicode because some older character encoding had a subset of Hebrew just for math notation, and the Unicode needs to take it into account. It's a complete mystery why the Unicode consortium didn't consult proper mathematicians, though. –  Yuji Sep 6 '11 at 13:35
    
@Yuji: Oh, I know that Hebrew is registered as Hebrew as well as into the mathematical notation. I just added that as a point of possible interest and humor :-) –  Asaf Karagila Sep 6 '11 at 14:03
    
I guessed you knew, but I explained it just in case:p From now on I'll use "transfinite cardinal" when referring to Hebrew alphabets. –  Yuji Sep 6 '11 at 14:14
    
If you want to see where $\aleph_0$ was born, follow this link, go to page 510 : 492 (it appears first on page 506 : 488, but page 492 is where it is properly defined). By the way Hausdorff in his Grundzüge der Mengenlehre uses $\aleph$ for the continuum. See e.g. this page. @Yuji: ping –  t.b. Sep 6 '11 at 18:46
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@Theo: It might interest you to know that one can directly link to a particular page in GDZ, like so. (Admittedly, you need to peer a bit at the HTML, so it might be inconvenient.) –  J. M. Sep 7 '11 at 2:08

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