Potential and actual infinity

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Current or actual infinity (late Latin actualis , “active”, “effective”) and potential or potential infinity (late Latin potentialis , “ according to the possibility or ability ”) denote two modalities of how the infinite can exist or be imagined. It is about the question of whether, first of all, an object area of ​​infinite power can actually exist in all its parts at a given point in time (realism with regard to actual infinity), or whether only certain elements exist or can be imagined or constructed (anti-realism with regard to actual infinity) Infinity, for example as constructivism ), so that only potential infinity can really exist. Second, if one accepts the principle possibility of actual infinity, the question is which objects are actually infinite. In the field of mathematical philosophy, the question of the real existence of infinitely powerful sets comes into consideration, including, for example, the class of natural numbers (which here presupposes a position that is also called “Platonism” with regard to mathematical objects). The anti-realistic (here mostly: constructivist) position could be formulated as “There is no greatest natural number, but a complete set of natural numbers does not exist” (potentially infinite).

In the history of philosophy and contemporary ontology , further candidates for actual infinite objects are discussed, among other things: an infinite amount of substances (e.g. atoms) or spatial and temporal units (especially as a continuum ), an infinite series of causes (the impossibility of which is one Prerequisite for many classical proofs of God ), as well as God .

Concept history

Anaximander introduces the concept of an infinite (a-peiros). Infinity is both limitless and indefinite.

The idea of ​​actual infinity can be derived from Plato's explanations. It is the definite form principle, the one that structures the material diversity of matter through limitation.

In the ontology of Aristotle the contrast between potentiality and actuality is fundamental and is also applied to sets of objects. Aristotle calls a set to which in principle an infinite number of objects can be added "potentially" infinite. From this he differentiates the concept of a set, which really already contains an infinite number of objects. According to Aristotle, this is impossible. With this, Aristotle also turns away from the fact that a certain, infinite principle comprehensively explains the unity of finite reality. According to him, “infinite” only refers to “that outside of which there is still something”.

This exclusion of an actual infinity is often used in ancient and medieval religious philosophy for evidence of the existence of God. Because this means that a progression that can in principle take an infinite number of steps can never be completed. That is why one considers an explainability of reality to be impracticable, which starts with certain objects, lists their causes, and so progresses in each case. Instead, God is accepted as the primary cause, which is not itself part of such a series of causes. For example with Thomas Aquinas .

Augustine , following Platonism, identifies God directly with the actual infinite.

The ancient and medieval ontological and religious-philosophical discussions often refer to these foundations.

At the transition to the Renaissance and early modern times, Nikolaus von Kues combined these traditions with mathematical problems. In numerous arithmetic and geometrical analogies, he tries to make it clear that it is impossible for the finite, discriminating understanding to grasp the actual unity of the infinite. An example of this is the impossibility of actually making straight and curved lines coincide with the progressive inscription of polygons with increasing number of edges in a circle. This circular quadrature problem had already found numerous treatments, including a. with Thomas Bradwardine . In recent research, the considerations of Cusanus are often compared with problems in the philosophy of mathematics, as they have been posed since the early representatives of mathematical constructivism, as well as with the considerations of Georg Cantor .

Cantor was of the opinion that the potentially infinite presupposes the actual infinite and thus a clear opponent of Johann Friedrich Herbart , who in turn saw the concept of the infinite as a changeable limit that can or must shift further at any moment.

Different conceptions in today's mathematics and philosophy of mathematics

The talk of infinite "sets", which has prevailed on the side of the actualists and has become the most important basis of mathematics in the form of axiomatic set theory, is criticized or rejected by the potentialists. In order to make the controversy of the concept of quantity clear, it is occasionally given in the following with quotation marks.

The simplest example of an infinite set is the set of natural numbers : You can specify a successor to every natural number , so there is no end. Each and every one of these numbers ( no matter how large) can be specified in full, but the set with each of its elements cannot.

From the standpoint of Finitisten is therefore as each other infinite range, not as a set-existent. But a finite set exists because it can be explicitly stated by specifying all its elements, such as, for example . In this sense, the “set” is only potentially infinite, since new elements can always be added to it, but it is never ready because not all of its elements can be written down.

Ultrafinitists raise the objection that even finite sets such as( n is any natural number) cannot be written down completely if n is so large that practical reasons prevent it - available paper, lifetime of the writer or number of elementary particles, which issafely below 10 100 in the observable part of the universe .

For the more moderate constructivist, on the other hand, a set is already given if there is an algorithm / procedure with which each element of this set can be constructed , i.e. specified, in a finite number of steps . In this sense, the set of natural numbers is actually infinite because it exists in the form of an algorithm with which every natural number can be generated in a finite number of steps. “Ready available” here is not the set as a summary of its elements, but only the algorithm, the operational rule , according to which it is gradually generated. Many constructivists therefore avoid the term “actual infinite” and prefer to designate sets such as those of the natural numbers as “operationally closed”, which simply means that the associated algorithm generates each element of the set sooner or later.

The range of real numbers is the classic case of a non-operationally closed set. An algorithm can only produce numbers that can be represented with a finite number of characters, and so it is possible to construct finite or countable sets of real numbers (for constructivists these are regular sequences of rational numbers ) (e.g. by simply adding every gives another name), but it is not possible to specify an algorithm that can produce any real number. Because he would have to be able to produce these in a countable number of steps, which is not possible because the set of real numbers is uncountable ( Cantor's second diagonal argument ). The “set” of real numbers cannot be specified by an algorithm (or finitely many), but one would need an infinite number of algorithms to generate all real numbers, and these infinitely many algorithms cannot in turn be generated by a higher-level algorithm (because from this it would also follow that the real numbers would have to be countable). The algorithms for generating all real numbers therefore do not form an operationally closed area, so they can hardly be called “ready-made” and therefore rather form a potential infinity.

Remarkably, despite these difficulties in generating the set of real numbers, the actual view of the infinity of real numbers can also be found occasionally on the constructivist side: the intuitionist Luitzen Egbertus Jan Brouwer sees the continuum as a primal intuition, i.e. something belonging to the human mind what is already given and in this sense actual infinite. However, the continuum is "something finished [...] only as a matrix, not as a set". Of course, Brouwer does not explain what the "matrix" means here.

In the philosophy of mathematics, there is, in addition to the rejection of all concepts of infinity (ultrafinitism), the exclusive acceptance of the potentially infinite (finitism), and beyond that the acceptance of the actual infinite only for operationally closed sets such as that of the natural numbers ( constructivism ), as well as acceptance the actual infinite only for the continuum (intuitionism), while Platonism accepts the actual infinite throughout.

Classical mathematics and at the same time the vast majority of today's mathematicians accept the actual infinite for all sets that can be defined on the basis of the axioms of Zermelo-Fraenkel set theory : the infinity axiom provides the existence of the set of natural numbers, the power set axiom that of the real numbers. On this axiomatic basis there is an infinite multitude of levels of actual infinity, which are characterized by different cardinal numbers. For the cardinal numbers, as for the real numbers, no general creation process can be specified that could produce all of them. Whether the “totality of all cardinal numbers” is a meaningful term, whether it can be understood as actual infinity, is also controversial among mathematicians. To understand this totality as a set in the sense of axiomatic set theory leads to a logical contradiction ( Cantor's first antinomy ).


Individual evidence

  1. a b Deiser, Oliver: Introduction to set theory, 2nd edition, Springer, Berlin 2004, ISBN 3-540-20401-6 , page 23
  2. Aristotle, Metaphysics ix, 6; Physics iii: "In general, the infinite only exists in the sense that something different and again another is taken, but what has just been taken is always finite, but always different and again different."
  3. Physics 3, 207a1
  4. ^ Thomas Aquinas, Summa contra gentiles i, 13
  5. De civitate Dei 12
  6. See Johannes Hoff: Contingency, contact, transgression , on the philosophical propaedeutic Christian mysticism according to Nikolaus von Kues, Freiburg - Munich: Alber 2007, ISBN 978-3-495-48270-4 . Jean-Michel Counet: Mathématiques et dialectique chez Nicolas de Cuse , Paris: Vrin 2000, ISBN 2-7116-1460-3 . Gregor Nickel: Nikolaus von Kues: On the possibility of mathematical theology and theological mathematics , in: Inigo Bocken, Harald Schwaetzer (ed.): Mirror and portrait . On the meaning of two central images in Nicolaus Cusanus's thinking. Maastricht 2005, 9-28; also in: Tübingen reports on functional analysis 13 (2004), 198-214. Jocelyne Sfez: L'hypothétique influence de Nicolas de Cues sur Georg Cantor dans la question de l'infinité mathématique , in: Friedrich Pukelsheim, Harald Schwaetzer (ed.): The understanding of mathematics of Nikolaus von Kues. Mathematical, scientific and philosophical-theological dimensions, communications and research contributions from the Cusanus Society 29, Trier 2005, 127-158.
  7. s. Brouwer 1975, p. 104