The differences between $\aleph$ and $\mathfrak{c}$ 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 :)
 A: *

*$\aleph_0$ 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.
A: 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.
