A ℶ-fixed point (beth-fixed point) is a cardinal that remains unchanged under the beth function. In other words, it is a cardinal κ satisfying the equation

ℶκ = κ.

Unlike the finite stages of the beth hierarchy, where each application of the powerset operation necessarily produces a strictly larger cardinal, a beth-fixed point represents a stage at which the transfinite indexing of the beth function and the resulting cardinal coincide. Such cardinals occupy an important position in higher cardinal arithmetic because they describe points of stability within an otherwise rapidly expanding hierarchy of powerset-generated cardinalities.

To understand beth-fixed points, it is first necessary to understand the beth hierarchy itself. The hierarchy begins with

ℶ₀ = ℵ₀,

and is defined recursively by

ℶα+1 = 2^ℶα

for successor ordinals, while for limit ordinals λ,

ℶλ = sup{ℶβ : β < λ}.

Each successor stage therefore represents another application of the powerset operation, while each limit stage collects together every preceding beth number. The beth hierarchy measures the cumulative growth produced by repeated exponentiation rather than by simple successor operations within the aleph hierarchy.

Since every successor beth number is strictly larger than its predecessor, one might expect the hierarchy to continue growing indefinitely without ever intersecting its own indexing system. Remarkably, however, this is not the case. The beth function is normal, meaning that it is both strictly increasing and continuous at limit ordinals. One of the fundamental properties of normal functions is that they possess arbitrarily large fixed points. Consequently, there exist infinitely many ordinals α satisfying

ℶα = α.

When such an ordinal is itself viewed as the initial ordinal of its corresponding cardinal, the resulting cardinal is called a beth-fixed point.

A beth-fixed point should not be interpreted as a cardinal for which taking another powerset leaves the cardinal unchanged. Cantor's Theorem proves that

2^κ > κ

for every cardinal κ.

Therefore, no cardinal can satisfy

2^κ = κ.

The fixed-point equation instead concerns the beth function as an indexed hierarchy, not the powerset operation itself. The equality

ℶκ = κ

means that after carrying out every earlier stage of the beth construction indexed below κ, the hierarchy arrives precisely at κ. The powerset operation itself continues to produce strictly larger cardinalities beyond that stage.

This distinction reflects one of the most subtle aspects of transfinite recursion. The beth hierarchy assigns a cardinal to every ordinal index. Ordinarily these indices and resulting cardinals differ significantly. At a beth-fixed point, however, the indexing process eventually "catches up" to the cardinal it produces. The ordinal used to label the stage and the cardinal generated by the hierarchy become identical. Such coincidences occur only at exceptionally large stages of the transfinite hierarchy and arise through the continuity properties of the beth function rather than through any collapse of powerset growth.

Beth-fixed points are closely related to the broader theory of fixed points of normal functions. Similar phenomena occur throughout ordinal analysis. The epsilon numbers satisfy

ω^α = α,

the zeta numbers are fixed points of the epsilon function, and the Veblen hierarchy systematically generates common fixed points of increasingly complex normal functions. Beth-fixed points belong to this same general tradition. Instead of studying exponentiation on ordinals, they investigate fixed points generated by repeated applications of the powerset operation as encoded by the beth hierarchy.

The relationship between beth-fixed points and the Generalized Continuum Hypothesis (GCH) is also significant. Under GCH,

ℶα = ℵα

for every ordinal α.

Consequently, every beth-fixed point is simultaneously an aleph-fixed point, since

ℵκ = κ

whenever κ is a beth-fixed point. Without GCH, however, the beth and aleph hierarchies generally diverge. In that setting, beth-fixed points and aleph-fixed points need not coincide, reflecting the independence of the continuum function from ordinary cardinal succession. Thus the study of beth-fixed points naturally intersects questions concerning cardinal exponentiation, forcing, and the behavior of the continuum.

Although beth-fixed points are enormous cardinals, they are not themselves classified as large cardinals in the modern technical sense. Their significance arises from the recursive behavior of the beth hierarchy rather than from elementary embeddings, reflection principles, ultrafilters, or other large-cardinal axioms. Nevertheless, they frequently appear in advanced investigations of cardinal arithmetic because they identify natural stages at which the cumulative powerset hierarchy exhibits a form of self-consistency with its own indexing process.

From a foundational perspective, beth-fixed points illustrate how transfinite recursion can eventually generate stable structural landmarks within endlessly expanding hierarchies. The beth function grows through repeated powerset operations, yet its continuity guarantees the existence of stages where the hierarchy and its indexing become synchronized. These fixed points therefore reveal that even the explosive growth of cardinal exponentiation possesses deep internal regularities governed by the general theory of normal functions.

Ultimately, a ℶ-fixed point is a cardinal κ satisfying ℶκ = κ, representing a stage where the cumulative beth hierarchy coincides with its own index. It does not signify a fixed point of the powerset operation itself, since Cantor's Theorem guarantees that every powerset is strictly larger than its underlying set. Instead, beth-fixed points arise from the recursive structure of the beth function and occupy important positions within higher cardinal arithmetic, transfinite recursion, and the general theory of fixed points of normal functions. They demonstrate that even among the vast hierarchy of powerset-generated infinities, stable structural milestones inevitably emerge, revealing another layer of order within the endlessly ascending landscape of transfinite mathematics.