1 Definition and basic properties
A successor ordinal is an ordinal that comes immediately after another ordinal in the well-ordered class of ordinals. It is the ordinal obtained by applying the successor operation once to a given ordinal. Successor ordinals are central to ordinal theory because they mark discrete steps in an otherwise transfinite hierarchy.
1.1 Ordinals and the successor operation
Ordinals measure positions in well-ordered sets rather than sizes alone. For any ordinal \(\alpha\), there is a next ordinal, usually written \(\alpha+1\). This operation extends the familiar notion of “adding one” from finite counting to the transfinite setting. The result is always a strictly larger ordinal.
1.2 Formal definition of a successor ordinal
An ordinal \(\beta\) is called a successor ordinal if there exists an ordinal \(\alpha\) such that \(\beta=\alpha+1\). In this case, \(\alpha\) is the predecessor of \(\beta\). Every successor ordinal has an immediate predecessor, unlike limit ordinals.
1.2.1 Predecessor ordinal
The predecessor of a successor ordinal is the unique ordinal whose successor it is. If \(\beta=\alpha+1\), then \(\alpha\) is the ordinal immediately before \(\beta\). This predecessor is determined uniquely by the ordinal structure.
1.2.2 Characterization as \(\alpha+1\)
Being of the form \(\alpha+1\) is the standard characterization of a successor ordinal. This description emphasizes that successor ordinals are formed by a one-step extension of an earlier ordinal. In contrast, limit ordinals are not reached by a single successor step.
1.3 Examples of successor ordinals
Successor ordinals appear at every stage of ordinal arithmetic. They include both ordinary finite numbers and many infinite ordinals.
1.3.1 Finite ordinals
The finite ordinals \(1,2,3,\dots\) are all successor ordinals. For example, \(3\) is the successor of \(2\), and \(n+1\) is the successor of \(n\). In the usual von Neumann representation, each finite ordinal is the set of all smaller finite ordinals.
1.3.2 Infinite successor ordinals
Many infinite ordinals are also successor ordinals. For instance, \(\omega+1\) is the successor of \(\omega\), and \(\omega^2+1\) is the successor of \(\omega^2\). These ordinals lie beyond the first limit ordinal \(\omega\) but still arise by a single successor step.
2 Relation to limit ordinals
Successor ordinals are one of the two main types of ordinals, the other being limit ordinals. The distinction is fundamental in transfinite arguments and in the classification of ordinal stages.
2.1 Distinguishing successor and limit ordinals
A successor ordinal has an immediate predecessor, while a limit ordinal does not. Limit ordinals are approached by smaller ordinals but are not obtained by adding one to any earlier ordinal. This dichotomy divides ordinal structure into stepwise and limiting behavior.
2.2 No ordinal strictly between an ordinal and its successor
There is no ordinal \(\gamma\) such that \(\alpha<\gamma<\alpha+1\). The successor \(\alpha+1\) is the least ordinal greater than \(\alpha\). This “gaplessness” is part of what makes ordinals into a well-ordered class.
2.3 Limit ordinals as non-successors
Every limit ordinal fails to be a successor ordinal. Such ordinals have no immediate predecessor and often represent accumulation points of earlier stages. Typical examples include \(\omega\), \(\omega\cdot 2\), and \(\omega^\omega\).
3 Arithmetic with successor ordinals
Successor ordinals interact with ordinal arithmetic in characteristic ways. Because ordinal operations are not commutative, the position of a successor can affect the outcome.
3.1 Successor under ordinal addition
Adding a successor on the right produces a further successor: \[ \alpha + (\beta+1) = (\alpha+\beta)+1. \] By contrast, adding on the left does not behave like ordinary arithmetic. For infinite ordinals, \(\!1+\omega=\omega\), while \(\omega+1\) is strictly larger than \(\omega\).
3.2 Successor under ordinal multiplication
Ordinal multiplication also preserves successor structure on the right: \[ \alpha\cdot(\beta+1)=\alpha\cdot\beta+\alpha. \] When \(\alpha\) is a successor or finite ordinal, the result often remains a successor, but the general behavior depends on the factors involved. The noncommutative nature of multiplication is especially visible in transfinite examples.
3.3 Successor under ordinal exponentiation
Exponentiation introduces further distinctions among successor and limit stages. For example, \(\alpha^{\beta+1}=\alpha^\beta\cdot\alpha\). Successor exponents often correspond to a final multiplicative step, whereas limit exponents are defined by taking suprema of earlier values.
4 Structural properties
Successor ordinals have several structural features that make them easy to identify within the ordinal hierarchy.
4.1 Immediate predecessor and uniqueness
A successor ordinal has exactly one immediate predecessor. If \(\beta\) is a successor, then there is a unique \(\alpha\) with \(\beta=\alpha+1\). This uniqueness follows from the well-ordering of ordinals and the definition of the successor operation.
4.2 Cofinality of successor ordinals
Successor ordinals have the smallest possible cofinal behavior among nonzero ordinals. Since they are reached after a single step from their predecessor, they do not require an unbounded increasing sequence of smaller ordinals to approach them.
4.2.1 Successor ordinals and cofinality one
The cofinality of a nonzero successor ordinal is \(1\). This reflects the fact that the ordinal is not a limit of a longer ascending sequence. The predecessor itself is already “close enough” in the ordinal order to determine the successor.
4.3 Well-ordering behavior
In a well-ordered class, every nonzero successor ordinal has an immediate predecessor, and every set of ordinals has a least upper bound. Successor ordinals thus function as discrete points in the continuum of ordinal stages. They are the ordinal analog of adjacent integers in the finite case.
5 Role in transfinite methods
Successor ordinals are essential in constructions that proceed by stages indexed by ordinals. They separate continuation steps from limit stages where aggregation occurs.
5.1 Transfinite induction
Transfinite induction proves a property for all ordinals by handling three cases: zero, successor, and limit. In the successor case, one assumes the property for \(\alpha\) and proves it for \(\alpha+1\). This step mirrors ordinary mathematical induction but extends it to all ordinal levels.
5.2 Transfinite recursion
Transfinite recursion defines objects one ordinal stage at a time. At a successor stage \(\alpha+1\), the definition uses the object already constructed at \(\alpha\). This allows step-by-step creation of sequences, hierarchies, and functions indexed by ordinals.
5.3 Successor stages in ordinal-indexed constructions
Many ordinal-indexed constructions alternate between successor and limit stages. At successor stages, a procedure is extended by one step; at limit stages, a union, supremum, or other aggregate is taken. The distinction is especially important in cumulative hierarchies and recursive definitions.
6 Examples and notation
Successor ordinals are commonly written in ways that make their predecessor explicit. Standard notation helps distinguish them from limit ordinals at a glance.
6.1 Standard notation for successors
The notation \(\alpha+1\) is the most common way to denote a successor ordinal. Sometimes ordinal notation systems use symbols that encode the predecessor and the final successor step. In informal discussion, “the successor of \(\alpha\)” is equivalent to \(\alpha+1\).
6.2 Concrete examples below and beyond \(\omega\)
Below \(\omega\), the successor ordinals are the finite ordinals greater than \(0\). Beyond \(\omega\), examples include \(\omega+1\), \(\omega+2\), \(\omega^2+1\), and \(\varepsilon_0+1\). Each of these is obtained by adding one to an earlier ordinal.
6.3 Common conventions in set theory
In set theory, ordinals are often identified with von Neumann ordinals, where each ordinal is the set of all smaller ordinals. Under this convention, a successor ordinal \(\alpha+1\) is the union \(\alpha\cup\{\alpha\}\). This representation makes the predecessor relationship transparent and supports recursive arguments on ordinals.