aleph numbers

Therefore only the aleph numbers can meaningfully be called regular or singular cardinals.

We can then define the aleph numbers as follows:

In mathematics, aleph numbers denote the cardinality (or size) of infinite sets.

The aleph numbers differ from the infinity ( ) commonly found in algebra and calculus.

The aleph numbers are indexed by ordinal numbers.

Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers.

In set theory, the Hebrew aleph glyph is used as the symbol to denote the aleph numbers, which represent the cardinality of infinite sets.

This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets).

Thus the regularity or singularity of most aleph numbers can be checked depending on whether the cardinal is a successor cardinal or a limit cardinal.

More important examples of normal functions are given by the aleph numbers which connect ordinal and cardinal numbers, and by the beth numbers .

More specifically, White Light uses an imaginary universe to elucidate the set theory concept of aleph numbers, which are more or less the idea that some infinities are bigger than others.

In addition to introducing Aleph numbers the authors cite Lewis Carrol's The Hunting of the Snark, where instructions are given to avoid boojums when snark hunting.

Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.