Absolute Infinity
Absolute Infinity
Cordelia Mühlenbeck, Helmut Schmidt University, Experimental Psychology Unit, Hamburg, Germany
Absolute Infinity as World, Mathematics, and Cognition
With the development of set theory by Georg Cantor, actual infinity was for the first time systematically, consistently, and operatively established within mathematics. Prior to Cantor, infinity in the Western tradition had predominantly been conceived as potential: from Aristotle to Kant it was understood as an endless progression or as a mere limiting idea, not as a completed given. Cantor revolutionized this understanding by conceiving the infinite, through actual infinity, as internally determinate and mutually comparable, thereby showing that there are different cardinalities of the infinite. Infinity thus became, for the first time, a legitimate mathematical object in its own right—and no longer merely a regulative idea or a residue of abstraction. However, Cantor distinguished the actual infinity of the world and of mathematics from that of the Absolute, the latter being the most powerful and the only one described as absolute infinity, and regarded as separate from the former two. The Absolute was therefore conceived as existing independently of world and mathematics.
Upon closer examination of absolute infinity, however, the actual infinity of the world and of mathematics (as well as that of the self and of our cognitive capacity) must be regarded as identical with absolute infinity itself; otherwise, absolute infinity—if excluded from a weaker infinite part—could no longer be absolutely infinite. In this way, space-time and the mathematical universe of all sets (the universal class) themselves become absolutely infinite, that is, every point within them is absolutely infinite. The same holds for our capacity to know both. How can this be? Is this not unthinkable and impossible? On the contrary, closer analysis shows that the initial assumption of any finitude (or complete structuration) of world and mathematics is itself impossible.
1. The Absolute Infinity of the World
The absolute infinity of the world can be justified ontologically, phenomenologically, and mathematically, since Being is not to be understood as one entity among others, but as the space-time continuum itself. Following Heidegger’s fundamental ontology, it becomes apparent that extensio—extension in space and time—constitutes the most fundamental commonality of Being and entities. Space and time are not separate containers in which things occur; rather, they coincide as space-time and constitute both the background and the concrete form of every entity. Things are not in space and time; they are themselves spatiotemporal contours within an encompassing continuum. This continuum is characterized by complete continuity: spatial points are not ultimate, discrete building blocks, but posited limit concepts within an existential space-time context. Every boundary itself possesses extension, such that an atomistic conception of space-time leads to contradictions. What follows is not merely infinite divisibility, but the infinity of the spatiotemporal itself. This insight is ontologically deepened by Heidegger’s determination of Being as becoming, in which Being and non-being belong together. Nothingness is not mere absence, but belongs, as refusal, to the essential unfolding (Wesung) of Being. It is precisely this indeterminacy that makes Being the totality of all possibilities. Since time, as the intuited movement of transition, always contains both the not-yet and the no-longer, it can never be completed. Space and time are therefore not finitely structurable, but carry an inner infinity within themselves. This infinity is not relative or transfinite, but absolute: it is not part of an order of cardinalities, but its very precondition. The spatiotemporal continuum is the ground from which all determinate domains of being can first emerge, without ever itself being exhausted.
This ontological diagnosis is corroborated by results from modern mathematics and logic. The identification of the continuum with the set of real numbers proves to be a projective simplification: real numbers are discrete elements, whereas the intuitive continuum is homogeneous. The undecidability of the continuum hypothesis, as well as Gödel’s incompleteness theorems, demonstrate that no formal system in the sense of classical mathematics can capture the continuum exhaustively. Every mathematical structure presupposes an infinite background that itself cannot be fully formalized. Mathematics thus points beyond itself toward an absolute infinity that underlies all formal infinities.
Kurt Gödel’s physical space-time models consistently reflect this idea of consequent ontological openness. His cosmological models (of the late 1940s), which incorporate the then-new insights of relativity theory, show that a global, linear flow of time cannot exist. Through rotating space-times, infinitely many equally legitimate timelines exist, such that the concept of an absolute time-line collapses. In his models space-time is not spatially closed or finitely bounded, but globally open and infinite. At every space-time point, infinitely many temporal forms and possibilities overlap. Thus, what is ontologically and mathematically implied is physically confirmed: every finite spatial or temporal interval contains the same absolute infinity as the whole.
From his analysis and description of space-time, and from his general results on formal systems, it follows that no physical theory can be both complete and closed with respect to the structure of the world. If space-time admitted a smallest unit or a final discrete structure, it would in principle allow a complete formal description; Gödel’s incompleteness results, however, rule out such total closure and thus undermine the idea of fundamental space-time discreteness. Consequently, any “world formula” or final theory that claims to exhaustively describe reality must be rejected, and quantum models that treat space-time as fundamentally discrete should be understood as effective, not ontologically final, descriptions. This conclusion does not follow from Gödel’s incompleteness theorems alone, but also from the ontological impossibility of discrete space-time points themselves: any purported point without extension would require a boundary that is itself extended, thereby presupposing a continuum and rendering fundamentally discrete points incoherent as ultimate constituents of reality.
The world is therefore not a finite universe within a larger framework, but itself the absolutely infinite space-time continuum. This absolute infinity is the fundamental-ontological condition of all existence, all possibility, and all knowledge. It is not relativizable, not enumerable, and not completable—but the inexhaustible ground from which all entities emerge.
This conception of absolute infinity also finds a classical metaphysical counterpart in Leibniz’s doctrine of the „Urmonad“. For Leibniz, no finite substance can exist by itself, since everything finite can be fully explained only through an infinite ground. The Urmonad—as a simple, non-composite, and absolutely infinite substance—thus functions as the necessary ground of all finite monads and their relations. It is not spatially localized, but contains within itself the totality of all possible perspectives of the world. Precisely because each individual monad reflects the universe in its own way, the world itself is intelligible only on the basis of an absolutely infinite ground. In a modern ontological reading, this Urmonad need not be understood as a personal subject, but as an expression of the absolute infinity of Being: as that inexhaustible background which—analogous to the space-time continuum—already contains all possible determinations, relations, and world trajectories within itself, without itself being finitely determined.
Since the physical description of the world is based on mathematical concepts, mathematical concepts and objects must, in some form, be real. That is, there exists a correspondence between mathematics and reality. We therefore deepen the examination of the infinities of mathematics in order to investigate more precisely the nature of absolute infinity, which has here already emerged in the ontological–physical argumentation.
2. The Absolute Infinity of Mathematics
Mathematics does not reveal itself merely as a formal system of finite rules, but as an expression of an absolute infinity that can neither be fully axiomatized nor exhaustively structured. This infinity does not concern only individual mathematical objects or transfinite cardinalities, but the fundamental constitution of mathematical concepts themselves. The starting point is the insight that mathematical concept formation does not proceed purely by convention or syntactic manipulation, but is grounded in an autonomous, non-empirical intuition. This intuition provides access to abstract structures that exist independently of the knowing subject and yet are effective within real orders, such as the physical world. Mathematics is therefore not merely a game with symbols, but a form of knowledge of an objective, conceptual reality.
This reality, however, is in principle unclosable. Every attempt to ground mathematics completely in a classical axiomatic system encounters internal limits. This becomes particularly clear in set theory, which functions as the foundation of modern mathematics. The distinction between sets and classes shows that there can be no “set of all sets.” Instead, the universe of all sets constitutes a proper class that eludes complete formal encapsulation. This structural openness is not a deficiency, but a necessary feature: if the mathematical universe as a whole were formally closable, it would no longer be infinite in the strict sense.
The current undecidability of fundamental questions—such as the continuum hypothesis—highlights this situation. The fact that neither its truth nor its falsity follows from the established axioms indicates not merely an epistemic limitation, but points to an ontological depth of mathematical reality itself. Different models of set theory realize different orders of infinity, without any one of them being definitively privileged. It follows that there is no final ordering of infinities, but rather a fundamentally open hierarchy that extends outward through ever larger cardinalities and inward through unlimited divisibility.
Decisive here is the distinction between discrete point structures and an underlying homogeneous continuum. The mathematical continuum in current use, identified with the real numbers, is treated as a discrete structure composed of points. Its necessary continuity—visible in the infinitely progressing processes of the digit expansions of every real number—however, depends on a presupposed, non-point-like background. Points are posited within this background, they are not its constitutive elements. Every formal set-theoretic structure therefore necessarily depends on an inexhaustible continuum that cannot be fully objectified. This continuum can meaningfully be understood only as absolute infinity.
General principles of maximality and reflection articulate this insight conceptually. They state that every structural determination of the mathematical universe can always be reflected into subdomains, while the universe as a whole remains indeterminate. Whenever one believes that “all” mathematical objects have been captured, the conceptual structure itself forces an extension. Mathematics therefore contains an intrinsic surplus: every one of its points, every one of its structures, refers beyond itself to the same absolutely infinite ground.
The absolute infinity of mathematics thus does not consist in the mere existence of very large infinities, but in the fact that mathematical totality as such can never be closed. This unclosability is not an external feature, but the inner condition of the possibility of mathematical knowledge itself.
3. The Absolute Infinity of Cognition and the Self
We now turn to the absolute infinity of our capacity for knowledge and of the self. The absolute infinity of the self follows necessarily from its ontological structure, provided the self is not understood as an isolated subject, but as a relation within Being. The self is not a finished thing or a closed unit, but arises and persists exclusively through relations: relations to the world, to itself, and—more fundamentally—to Being as such. In this way, the self shares the ontological constitution of that to which it relates. If Being itself is determined as an absolutely infinite spatiotemporal continuum, then the self cannot be finite in a strict sense, but is essentially open toward absolute infinity.
This infinity of the self follows initially from the fact that every act of self-determination necessarily points beyond itself. The self can determine itself only by distinguishing itself—from the world, from others, from earlier or possible states of itself. Yet every such determination presupposes a horizon within which distinction is possible at all. This horizon cannot itself be finitely fixed, since every boundary implies a further horizon. The self is therefore structurally and irreducibly horizontal: it always already stands within an open field of possibilities, meanings, and world-relations that cannot be closed in principle.
This can be specified more precisely at the fundamental ontological level. The self, like any entity, is determined by its Being. Being, however, is not an object among others, but the comprehensive process of spatiotemporal becoming itself. Since this spatiotemporal continuum is absolutely infinite, every self-structure situated within it is necessarily constituted together with this infinity. The self is not in space-time like a point in a container, but a spatiotemporal contour, a relational zone of condensation within the infinite continuum. It has no fixed boundary, but continuously shifts and transforms its contours through experience, reflection, and engagement with the world.
This ontological openness manifests phenomenologically in the unclosability of self-reflection. No self can fully grasp itself, since every act of reflection generates a new reflexive relation. The self becomes aware of itself by making itself an object, thereby necessarily producing a new self-relation that is itself unclosed. This infinite regress is not a logical defect, but an expression of the absolute infinity of the self. If the self were finitely determinable, there would have to be a final, fully transparent concept of the self—but such a concept does not exist in principle.
From a developmental-ontological perspective, this infinity appears as an inner orientation. Development is not the progression through predetermined stages toward a final goal, but a continuous process of differentiation within an open horizon. The self develops by forming new relations to the world, to others, and to itself. These relations are never final, since they always refer to a more comprehensive space of meaning and possibility. This space of possibility is identical with the absolute infinity of Being itself, in which the self necessarily participates.
The continuous formation of new relations and the ongoing shaping of the self-concept correspond to the figure–ground segregation that underlies all concept formation. The figure–ground segregation constitutes the essence of concept formation by allowing a specific content to emerge as a figure from an implicit background that cannot be fully explicated, thereby making determinacy possible at all. Every concept is thus constituted relationally: it is intelligible only through the ground from which it stands out, while this ground itself remains an open context that cannot be fully thematized.
The self therefore reveals itself as a finite form of appearance with an infinite underlying structure. It is finite in its concrete life processes, experiences, and decisions, but infinite in its ontological openness, its reflexivity, and its fundamental orientation toward Being. The absolute infinity of the self is thus neither a psychological attribution nor a metaphysical ideal, but the necessary condition of the possibility of self-consciousness, development, and freedom as such. The self is never finished, never fully determined, and never complete—precisely because, in its essence, it is oriented toward absolute infinity.
Since this absolute infinity of the self, as described, corresponds to the general openness of concept formation, it is also reflected in our capacity for knowledge. The human ability to grasp absolute infinity does not manifest itself in the complete apprehension of a closed object, but in the formation of open concepts that reflect their own principled unclosability. Our concept formation is structured in such a way that it can transcend any formal fixation by making the conditions of its own application into an object of reflection, thereby continuously opening new levels of abstraction. This structure is not limited to mathematics, but applies to cognition in general. Absolute infinity is not possessed as a totality, but recognized as a conceptual openness that sustains every determination while simultaneously exceeding it. In this sense, absolute infinity does not appear as a limit of knowledge, but as its constitutive condition.
Across all three domains we find the same fundamental relation: the existential relation between entities and their Being in the world (ontology, cosmology), between figure and ground in cognition, between points and domains in mathematics, and between self- and world-concepts and the absolutely infinite self- and world-continuum in cognitive development. In every finite act of cognition and every finite mode of existence, absolute infinity is necessarily co-given as an integral component of the formation of finite figures, and is thereby implicitly recognized and present.
Across world, mathematics, and cognition, the same structural feature emerges: absolute infinity is not an external limit or a highest object, but the underlying condition that makes finite structures, distinctions, and determinations possible at all. Space-time, the mathematical universe, and the self are not merely infinite in different ways; they are identical expressions of one and the same absolute infinity, articulated on ontological, formal, and cognitive levels. This identity is revealed in the unclosability of space-time, the openness of mathematical totality, and the irreducible reflexivity of cognition. Absolute infinity therefore does not stand beyond world, mathematics, and cognition, but is precisely what they are, insofar as they can appear, be structured, and be known.
Cordelia Mühlenbeck was awarded the Kurt Gödel Prize 2025 First Prize in the essay competition “How are Gödel’s conceptual and mathematical realism, his argument against the existence of time, and his ontological argument compatible with a coherent ontology?” Her education and academic trajectory include a B.Sc. in Mathematics, a B.A. in Multilingual Communication, and an M.A. in International Art Management, followed by a Dr. phil. (PhD) in Psychology with a specialization in Evolutionary Psychology. She is currently pursuing a habilitation in Developmental Psychology, focusing on a meta-theory of developmental psychology. She has extensive experience in university teaching, particularly in psychological methodology and philosophy of science. Her research interests span several disciplines: in psychology, she works in evolutionary psychology, developmental psychology, and experimental psychology; in philosophy, her focus lies on metaphysics and ontology as well as the philosophy of mathematics; and in mathematics, her research centers on foundational questions concerning the relationship between the meta-mathematical conceptual level and the formal level.
