Aleph (ℵ) in LaTeX
\aleph produces ℵ, the Hebrew letter aleph, used in set theory (always with a subscript) to name the cardinalities, or sizes, of infinite sets.
LaTeX command
\alephNot the symbol you meant? Draw it to find the command →
\aleph_0, aleph-null, is the cardinality of the natural numbers, the smallest infinite cardinal, and every countably infinite set (the integers, the rationals) shares this same cardinality. Larger infinite cardinals are indexed \aleph_1, \aleph_2, and so on, forming the aleph hierarchy.
This is distinct from the cardinality of the real numbers, written \mathfrak{c} or 2^{\aleph_0}, which is strictly larger than \aleph_0. Whether \mathfrak{c} equals \aleph_1 is exactly the continuum hypothesis, a famous statement shown to be independent of the standard ZFC axioms of set theory.
\aleph is essentially never written bare; it needs a subscript specifying which cardinal is meant. Cardinal numbers (aleph) should also not be confused with ordinal numbers, which index a different, related hierarchy and are usually written with \omega instead.
Examples
$|\mathbb{N}| = \aleph_0$the cardinality of the naturals, aleph-null
$\aleph_0 < \aleph_1 < \aleph_2 < \cdots$the aleph hierarchy of infinite cardinals
$2^{\aleph_0} = \mathfrak{c}$the cardinality of the continuum
Tips and common mistakes
- Always attach a subscript,
\aleph_0,\aleph_1, and so on; a bare\alephwith no index is not standard usage. \alephneeds no extra package in LaTeX or KaTeX.- Don't confuse cardinal aleph numbers with ordinal numbers; ordinals index a related but different hierarchy, conventionally written using
\omega. - The continuum hypothesis, 2^{
\aleph_0} =\aleph_1, is independent of ZFC, worth a footnote if you state it as if it were simply true or false.
Working from handwritten notes?
LatexNotes turns a photo or PDF of handwritten mathematics into clean, compilable LaTeX - graphs and all. Free to try, no sign-up.
Convert your maths to LaTeX →FAQ
What does \aleph_0 mean?
Aleph-null, the cardinality of the natural numbers, the smallest infinite cardinal number.
How do I write aleph in LaTeX?
\aleph, then add a subscript for the specific cardinal, e.g. \aleph_0 or \aleph_1.
Is \aleph the same as infinity (\infty)?
No. \infty is a single symbol for an unbounded limit or an extended real value. \aleph_0, \aleph_1, and so on are a family of distinct infinite cardinal numbers from set theory.
What is the continuum hypothesis?
The claim that there is no cardinal strictly between \aleph_0 and the cardinality of the real numbers, i.e. that 2^{\aleph_0} = \aleph_1; it is independent of the standard ZFC axioms.