What are the Aleph numbers?

In the realm of mathematics, Aleph numbers play a significant role, serving as a series of numbers that denote the cardinality or size of infinitely large, well-ordered sets. The smallest Aleph number, aleph-zero (ℵ0), symbolizes the cardinality of natural numbers and is demonstrated by several sets such as integers, rational numbers, constructible numbers, algebraic numbers, computable numbers, finite binary strings, and finite subsets of any given countably infinite set.

The progression from aleph-zero leads to the next larger cardinality, aleph-one (ℵ1), and continues in a similar pattern to define a cardinal number ℵα. It’s noteworthy that the cardinality of infinite ordinal numbers is represented by an Aleph number, and every Aleph corresponds to the cardinality of a certain ordinal, making it well-orderable.

While finite sets are well-orderable, they do not possess an Aleph number as their cardinality. The conjecture that the cardinality of each infinite set corresponds to an Aleph number is akin to the axiom of choice, implying the existence of well-ordering for every set.

In the context of ZFC set theory that includes the axiom of choice, it’s inferred that the cardinality of every infinite set is an Aleph number. However, without the axiom of choice in ZF set theory, the proof that each infinite set has an Aleph number as its cardinality becomes unattainable. Scott’s trick is then employed as an alternate method to construct representatives for cardinal numbers.