Aleph1

Aleph1, denoted aleph_1, is the first uncountable cardinal in set theory. It is the cardinality of ω1, the first uncountable ordinal, and therefore the initial cardinal after aleph_0. In ZFC, aleph_1 is the successor cardinal of aleph_0; every countable set has cardinality at most aleph_0, and any set with cardinality aleph_1 is uncountable.

The concept arises from the axiom of choice, which ensures that every set can be well ordered, producing a well-defined sequence of alephs. aleph_1 is the smallest cardinal that is strictly larger than aleph_0, serving as a baseline for comparing sizes of infinite sets beyond the countable.

The relationship to the real line is given by the continuum c = |R|. In ZFC one has aleph_1 ≤ c. The Continuum Hypothesis (CH) asserts c = aleph_1, meaning the reals have the smallest possible uncountable cardinality. Without CH, the continuum can be strictly larger than aleph_1; in many models of set theory, c may be aleph_2 or larger.

In frameworks without the axiom of choice, the situation is more nuanced: ω1 remains the first uncountable ordinal, and aleph_1 is its cardinal, but not all sets can be well ordered in general, which affects the status of the alephs.

Aleph_1 plays a central role in cardinal arithmetic, descriptive set theory, and the study of ordinal structure, serving as the baseline for comparing sizes of infinite sets beyond the countable.