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In elementary set theory, the cardinal of the real numbers is denoted by $|\mathbb{R}| = \aleph$. After a few months here on MathExchange, I have seen quite a few times the notation $\mathfrak{c}$ for the exact same thing. Recently an answer of mine was edited such that every time I used $\aleph$ it was replaced with $\mathfrak{c}$ which made me wonder: Are there any actual differences between the two notation? Of course, they're defined to be the exact same thing, but maybe differences in the source of the notation, or where you should use each, or something like that...

I appreciate your help in advance :)

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    $\begingroup$ en.wikipedia.org/wiki/Aleph_number seems to imply that we define $\aleph_0=|\Bbb N|$ but it does not define $|\Bbb R|$ using aleph-notation directly. Instead it recites the continuum hypothesis that $|\Bbb R|=\aleph_1$ is the next largest infinity, which is not provable. I have never personally seen $\aleph$ used for the cardinality of the real numbers. The wiki page on cardinality of the continuum follows the same naming structure as the earlier article. $\endgroup$
    – JMoravitz
    May 16 '17 at 17:38
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    $\begingroup$ Yes, in Israel $\aleph$ is used in some places to denote the cardinality of the continuum. This is somewhat less common internationally. $\endgroup$
    – Asaf Karagila
    May 16 '17 at 17:55
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    $\begingroup$ See math.stackexchange.com/questions/444273/… (specifically the comments) for example. I am fairly sure this was discussed on the site before, but I can't find anything right now. $\endgroup$
    – Asaf Karagila
    May 16 '17 at 17:58
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    $\begingroup$ I found that question I was looking for. Maybe now people will stop posting irrelevant answers... :P $\endgroup$
    – Asaf Karagila
    May 16 '17 at 18:03
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    $\begingroup$ Sorry but the irrelevance of our answers was not apparent until we had the unexpected information that you had been reading about the subject in Hebrew. $\endgroup$
    – badjohn
    May 16 '17 at 18:23
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  • $\aleph_0$ is the cardinality of the set $\{0,1,2,3,\ldots\}$ of all finite cardinalities. These are linearly ordered in a way that gives each of them only finitely many predecessors.
  • $\aleph_1$ is the cardinality of the set of all countable ordinals.
  • $\frak{c}$ is the same as $2^{\aleph_0}$ and is the cardinality of the set of all real numbers. The notation $a^b$ means the cardinality of the set of all mappings from a set of size $b$ into a set of size $a$. Hence $2^{\aleph_0}.$

These notations were introduced by Georg Cantor in the 19th century. Cantor proved that $\aleph_0<\aleph_1$ and that $\aleph_0 < 2^{\aleph_0}.$ He showed that $\aleph_1 \le 2^{\aleph_0}$ using (what would later be recognized as) the axiom of choice, and he conjectured that those are equal. Much later it was shown that standard axioms of set theory do not give enough information to determine whether they are equal. In set theory without the axiom of choice is is possible that neither is greater than the other but they are nonetheless not equal.

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    $\begingroup$ This is entirely unrelated to the question which asks whether or not $\aleph$ and $\frak c$ denote the same object. $\endgroup$
    – Asaf Karagila
    May 16 '17 at 17:58
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    $\begingroup$ @AsafKaragila : "entirely unrelated" is a considerable exaggeration. $\endgroup$ May 16 '17 at 18:16
  • $\begingroup$ I don't know about considerable. (I did not downvote, though.) $\endgroup$
    – Asaf Karagila
    May 16 '17 at 18:23
  • $\begingroup$ The answers are not entirely irrelevant since they demonstrate how the use of $\aleph$ in the sense of this question can be misunderstood, which explains why that notation is avoided. $\endgroup$ Nov 17 at 12:27
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There is an important difference. $\aleph$ refers to a series of infinite cardinals and is usually written with a suffix. The smallest is $\aleph_0$ which is defined to be the cardinality of the natural numbers. $\aleph_1$ is the next biggest infinity. The cardinality of the reals is provably greater than $\aleph_0$ and hence at least as big as $\aleph_1$. Whether it is the same is The Continuum Hypothesis. Look that up. You might also want to look at beth $\beth$ numbers.

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  • $\begingroup$ Again, as I wrote under Michael's answer, this has nothing to do with the question at hand. Which is purely notational. $\endgroup$
    – Asaf Karagila
    May 16 '17 at 18:03
  • $\begingroup$ You asked whether there was a difference. So, we said that there was. My apologies for misunderstanding. $\endgroup$
    – badjohn
    May 16 '17 at 18:26
  • $\begingroup$ There is more than one "Asaf" on the site. $\endgroup$
    – Asaf Karagila
    May 16 '17 at 18:27
  • $\begingroup$ Another apology, as a John, I should have been more aware of that possibility. The Hebrew connection was a surprise though. $\endgroup$
    – badjohn
    May 16 '17 at 18:37

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