In set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets.They are named after the symbol used to denote them, the Hebrew letter aleph ().. He is the youngest character in Infinio's family. Stack Exchange Network. The symbol aleph_0 is often pronounced "aleph-null" rather than "aleph-zero," probably because Null is the word for "zero" in Georg Cantor's native language of German. Aleph-null definition is - the number of elements in the set of all integers which is the smallest transfinite cardinal number. Or would it be a wrong thing to do? Aleph null (also Aleph naught or Aleph 0) is the smallest infinite number. The cardinality of the natural numbers is (read aleph-naught; also aleph-null or aleph-zero), the next larger cardinality is aleph-one , then and so on. Aleph-null is by definition the cardinality of the set of all natural numbers, and (assuming, as usual, the axiom of choice) is the smallest of all infinite cardinalities.A set has cardinality if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Continuing in this manner, it is possible to define a cardinal number aleph_alpha for every ordinal number α, as described below. To me it seems intuitively obvious that yes, pi (and any other irrational number) would have to have aleph null number of digits--if it were higher cardinality asking about even, say, the 3rd digit of pi wouldn't make sense. Aleph-null. (aleph-naught, also aleph-zero or the German term Aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called ω or ω0 (where ω is the lowercase Greek letter omega), has cardinality . Aleph-null definition is - the number of elements in the set of all integers which is the smallest transfinite cardinal number. $\begingroup$ Sure, I know that (the quote-marks were meant to indicate that aleph-zero-minus-one isn't a real thing). The cardinality of the natural number s is aleph_0 (aleph-null, also aleph-naught or aleph-zero), the next larger cardinality is aleph-one aleph_1, then aleph_2 and so on. The set theory symbol aleph_0 refers to a set having the same cardinal number as the "small" infinite set of integers. I've been thinking about this after watching Vi Hart's videos about the types of infinities. It is the cardinality (size) of the set of natural numbers (there are aleph null natural numbers). Would I be right in saying that $\aleph_0 \in \mathbb N$? He was born in Germany.