Cardinals, infinite cardinals, regular cardinals, and successor cardinals belong to the foundational language of set theory used to classify the sizes of collections and the structural behavior of transfinite magnitudes. A cardinal number measures how many elements a set contains without regard to the order in which those elements are arranged. Two sets possess the same cardinality whenever there exists a one-to-one correspondence between their members. This principle allows cardinality to generalize ordinary counting beyond finite collections and into the transfinite hierarchy, where different kinds of infinity can be rigorously distinguished from one another.

A cardinal is therefore an abstract measure of size. For finite sets, the cardinals are the familiar natural numbers 0, 1, 2, 3, and so forth. A set containing five elements has cardinality 5 regardless of what those elements are or how they are arranged. In set theory, cardinals are commonly represented by initial ordinals: an ordinal κ is a cardinal when no smaller ordinal has the same cardinality as κ. This convention allows cardinal numbers to be embedded directly into the ordinal hierarchy while preserving their role as representatives of distinct set sizes.

An infinite cardinal is a cardinal greater than every finite cardinal. The smallest infinite cardinal is ℵ₀, the cardinality of the natural numbers. Beyond ℵ₀ comes an ascending sequence of larger cardinals,

ℵ₀, ℵ₁, ℵ₂, ℵ₃, ...

where each aleph denotes a distinct infinite cardinality. The notation continues through transfinite indices such as ℵω, ℵω+1, and far beyond. This aleph hierarchy demonstrates that infinity is not represented by a single magnitude. There are endlessly many distinct cardinalities, each strictly larger than those preceding it.

The difference between finite and infinite cardinal arithmetic is profound. For finite cardinals, adding or multiplying positive quantities generally increases their magnitude. Infinite cardinals frequently behave differently. For example,

ℵ₀ + ℵ₀ = ℵ₀

and

ℵ₀ · ℵ₀ = ℵ₀.

A countably infinite collection can therefore be combined with another countably infinite collection without producing a larger cardinality. Similar behavior occurs for every infinite cardinal κ under ordinary cardinal addition and multiplication:

κ + κ = κ

and

κ · κ = κ.

Cardinal exponentiation, however, is substantially more complicated and produces some of the deepest questions in set theory.

Among infinite cardinals, an essential distinction is made between regular and singular cardinals. This distinction is determined by cofinality. The cofinality cf(κ) of an ordinal or cardinal κ is the smallest order type of an unbounded subset of κ. Informally, it measures the shortest possible increasing sequence required to approach κ from below. A cardinal κ is called regular when

cf(κ) = κ.

This means that κ cannot be reached as the supremum of fewer than κ smaller ordinals.

The smallest infinite cardinal ℵ₀ is regular because no finite sequence of natural numbers can be unbounded in ω. Reaching the entirety of the natural numbers requires a countably long sequence. Likewise, ℵ₁ is regular in ZFC: no countable sequence of countable ordinals can be cofinal in ω₁. Every countable collection of ordinals below ω₁ has a countable supremum and therefore remains below ω₁ itself. Regularity consequently describes a kind of structural resistance to being assembled from substantially smaller pieces.

A singular cardinal, by contrast, is an infinite cardinal κ satisfying

cf(κ) < κ.

Such a cardinal can be approached by a sequence whose length is strictly smaller than the cardinal itself. A standard example is ℵω. Since

ℵω = sup{ℵ₀, ℵ₁, ℵ₂, ...},

the sequence of smaller cardinals indexed by the natural numbers is cofinal in ℵω. Consequently,

cf(ℵω) = ω,

and ℵω is singular. This distinction becomes extraordinarily important in advanced cardinal arithmetic, especially through results such as König's theorem, the Singular Cardinals Hypothesis, and Shelah's pcf theory.

Another major classification is the distinction between successor cardinals and limit cardinals. Given any cardinal κ, its successor cardinal, written

κ⁺,

is the smallest cardinal strictly greater than κ. Thus,

ℵ₀⁺ = ℵ₁,

ℵ₁⁺ = ℵ₂,

and more generally,

ℵα⁺ = ℵα+1.

A successor cardinal does not mean the ordinal successor κ + 1. Cardinal succession asks for the next distinct cardinal magnitude, whereas ordinal succession merely appends one new position to an ordering. For an infinite cardinal κ, the ordinal κ + 1 has exactly the same cardinality as κ and therefore is not the successor cardinal κ⁺.

This distinction between κ + 1 and κ⁺ is fundamental. If κ is infinite, then adding a single element does not increase its cardinality:

|κ + 1| = κ.

By contrast, κ⁺ is, by definition, a genuinely larger cardinal for which no intermediate cardinal exists. Thus ℵ₁ is not obtained merely by adding one element to ℵ₀; it represents the least possible cardinality that strictly exceeds countability. Successor cardinals therefore mark discrete transitions within the hierarchy of cardinal magnitudes.

Under the Axiom of Choice, every infinite successor cardinal is regular or may it be singular? The precise statement requires care: successor cardinals are always regular is not provable in ZFC and is false in general. What ZFC guarantees is that the cofinality of a successor cardinal cannot be too small in certain ways, but successor cardinals can consistently be singular in the absence of Choice, while under ZFC successor cardinals are indeed regular? This common simplification must itself be corrected: ZFC proves that successor cardinals need not generally be inaccessible-type regular objects solely by being successors; however, the standard theorem is that every successor cardinal is regular under the Generalized Continuum Hypothesis, not from successorhood alone. Thus regularity and successorhood are logically distinct properties and should never be treated as interchangeable classifications.

A limit cardinal is an infinite cardinal that is not the successor of any cardinal. Within the aleph hierarchy, ℵω is the canonical example because there is no single aleph immediately preceding it whose successor is ℵω. Instead, ℵω is the supremum of the sequence

ℵ₀, ℵ₁, ℵ₂, ...

Limit cardinals can themselves be either regular or singular. When an uncountable cardinal is both regular and a strong limit cardinal, one reaches the definition of a strongly inaccessible cardinal, one of the first major large-cardinal concepts. Thus the elementary distinctions among cardinality, regularity, succession, and limit behavior eventually become foundational ingredients in the large-cardinal hierarchy.

The concept of a strong limit cardinal further illustrates why these classifications matter. An infinite cardinal κ is a strong limit cardinal when

2^λ < κ

for every cardinal λ < κ.

If κ is simultaneously uncountable, regular, and a strong limit, then κ is strongly inaccessible. This reveals how properties introduced at the elementary level combine to generate substantially stronger set-theoretic notions. Regularity controls how κ can be approached from below, while the strong-limit condition controls how rapidly powersets below κ can grow. Together they produce a cardinal possessing exceptionally strong closure properties.

Cardinals therefore possess multiple independent structural characteristics. A cardinal may be finite or infinite, regular or singular, successor or limit, strong limit or non-strong-limit, and may satisfy increasingly powerful additional principles. These labels do not merely describe how “large” a cardinal is. They identify fundamentally different structural behaviors. Two cardinals can stand very far apart in size while sharing regularity, while nearby cardinals can differ dramatically in cofinality, powerset behavior, or reflection properties.

Ultimately, cardinals provide the fundamental language for measuring set-theoretic magnitude; infinite cardinals extend this measurement into an endlessly ascending hierarchy of distinct infinities; regular cardinals are those whose cofinality equals their own cardinality; and successor cardinals are the immediate next cardinal magnitudes above preceding cardinals. Together these concepts establish the structural grammar from which much of advanced set theory develops. Before inaccessible cardinals, measurable cardinals, supercompact cardinals, elementary embeddings, and stronger principles can be meaningfully understood, the underlying distinctions among size, cofinality, succession, and limit behavior must first be established. These apparently elementary ideas are therefore not minor preliminaries, but some of the essential foundations upon which the broader architecture of transfinite set theory is constructed.