Actual infinity
Actual infinity ( late latin actualis, " active ", " active" ) and potential or potential infinity ( late latin potentialis, " the possibility or the assets after " ) denote two modalities, such as infinite exist or can be imagined. It is about the question, first, whether any one subject area of infinite thickness in all its parts may actually exist at a given time (realism regarding aktualer infinity ), or may be presented or constructed ( anti-realism with respect to only one certain elements exist or aktualer infinity, such as constructivism), so only potential infinity can actually exist. Second, there is, one accepts the possibility in principle aktualer infinity to the question of which objects are actually infinitely. In the field of philosophy of mathematics this is particularly the question of the real existence of infinitely more powerful quantities into account, including, for example, the class of natural numbers ( which presupposes a position here, which is also " Platonism " with respect to mathematical objects called ). The anti-realistic ( mostly here: constructivist ) position could be formulated as " While there is no largest natural number, but a finished totality of natural numbers does not exist" (potentially infinite).
In the history of philosophy and the current ontology are discussed as additional candidates for aktual infinite objects, among other things: an infinite amount of substances (such as atoms ) or to spatial and temporal units (especially as a continuum ), an infinite series of causes ( whose impossibility is a requirement of many classical proofs of God ), and God.
Conceptual history
Anaximander introduces the notion of an unbounded (a- peiros ). Infinity is equally limitless as indeterminate.
Plato's explanations can be found in the idea of an actual infinity. It is the particular form principle, the One, which structures the material diversity of matter by narrowing.
In the ontology of Aristotle the contrast between potentiality and actuality is fundamental and is also applied to sets of objects. A lot, which in principle infinitely many objects are addable, Aristotle calls " potentially " infinite. Of this he distinguishes the notion of a quantity which actually already contains infinitely many objects. This is impossible according to Aristotle. Aristotle thus turns also from the fact that a certain, infinite principle explains the unity of the finite reality comprehensively. " Infinity " refers, according to him only to " the one outside it is still something."
This exclusion of an actual infinity is often used in ancient and medieval philosophy of religion for evidence of the existence of God. Because that is a progression that can in principle take place infinitely many steps, never locked. That's why you're making a explicability reality impracticable that start at certain objects, cites their causes, and so progresses, respectively. Instead, God is assumed to be first cause, which is not element of such a cause series itself. For example, in Thomas Aquinas.
Augustine identified the following Platonism, God directly aktualen with the Infinite.
The ancient and medieval ontological and religious-philosophical discussion refer frequently to these basics.
At the transition to the Renaissance and Early Modern Nicholas of Cusa combines these traditions with mathematical problems. In numerous arithmetic and geometric analogies he tries to clarify that it is the finite, distinctive sense impossible to detect the aktuale unity of the infinite. An example of this is the impossibility to bring by progressive Einbeschreibung of polygons with an increasing number of edges in a circle straight and curved line aktual to cover. This problem of squaring the circle had already found many treatments, including Thomas Bradwardine. In the recent research often the considerations of Cusa with problems of the philosophy of mathematics are compared, as they stand since the early representatives of a mathematical constructivism, as well as the considerations Georg Cantor.
Cantor was of the opinion that the potential infinity presupposes the actual infinity and thus a clear opponent of Johann Friedrich Herbart, who in turn looked at the concept of infinity as convertible limit which at any moment may continue to shift and needs.
Different conceptions in today's mathematics and philosophy of mathematics
The speech of infinite " sets", which has established itself on the side of Aktualisten and has become, in the form of axiomatic set theory to the most important foundation of mathematics, criticized or rejected by the Potentialisten. To make the contentiousness of the quantity concept clearly, he is sometimes provided in the following with quotation marks.
The simplest example of an infinite set is the set of natural numbers: For every natural number can specify a successor, so there's no end. Each of these numbers ( and be it ever so great ) can be specified completely, the amount of each of its elements is not.
From the standpoint of Finitisten is therefore, like any other infinite area, non-existent as a set. A finite set exists but, as it can by specifying all of its elements, as for example, be explicitly stated. The " amount" in this sense is only potentially infinite, since you brought new elements may be added forever, but they never finished exists because not all of its elements can be written down.
Ultrafinitisten raise here the objection that even finite sets such as ( n is an arbitrary natural number) can not be completely written down, if n is so large that for practical reasons prevent this - that becomes available paper, life of the writer or the number of elementary particles, the safe is below 10100 in the accessible part of the universe.
For the more moderate constructivists, however, a lot has already been given if there is an algorithm / method which constructs each element of this set in finite number of steps, so it can be specified. The set of natural numbers is actually infinitely in this sense, because it exists in the form of an algorithm with which one can produce any natural number in a finite number of steps. "Done in the present " is here but not the amount as a summary of its elements, but only the algorithm, the operation rule under which it is gradually generated. Therefore, many constructivists avoid the term " actually infinitely " and describe how the quantities of natural numbers rather " complete operational " as what is to simply say that the associated algorithm generates each element of the set sooner or later.
The field of real numbers is the classic case of a non-operative closed set. An algorithm can produce only numbers that can be represented with a finite number of characters, and so it is possible, finite or countable sets of real numbers ( for constructivists are regular sequences of rational numbers) to construct ( by, for example, simply each Specifies a name ), but it is not possible to provide an algorithm that can generate any real number. Because of these should be able to produce in countably many steps, but this is not possible because the set of real numbers is uncountable ( Cantor's second diagonal argument). The " amount" of the real numbers, so can not by an algorithm (or finitely many ) are given, but one would need an infinite number of algorithms to generate all real numbers, and these infinitely many algorithms can not turn by a higher-level algorithm produce ( because also it would follow that the real numbers would be countable ). The algorithms for the generation of all real numbers therefore do not form a surgically closed area, and are therefore difficult to " present ready " to be called and therefore form more of a potential infinity.
Remarkably, despite these difficulties, the set of real numbers to produce sporadically encountered with respect to the infinity of the real numbers on the constructivist side of the aktuale view: The intuitionist Luitzen Egbertus January Brouwer sees the continuum as an original intuition, that is, as something the human mind ready given and thus aktual infinite. As such, the set of real numbers is the common mathematical modeling of the continuum, and they can then be conceived as infinite -update.
There is thus in the philosophy of mathematics in addition to the rejection of all infinity terms ( Ultrafinitismus ) the exclusive acceptance of the potentially infinite ( finitism ), beyond going the acceptance of the actual infinite of only surgically closed sets as the natural numbers ( constructivism), and the acceptance actual infinite only for the continuum ( intuitionism ), while Platonism actual infinity continuously accepted.
The classical mathematics, while the vast majority of today's mathematician accepts actual infinity for all quantities can be defined on the basis of the axioms of Zermelo -Fraenkel set theory: the axiom of infinity yields the existence of the set of natural numbers, the power set axiom that the real numbers. On this axiomatic basis results in an infinite variety of levels of actual infinity, which are characterized by different cardinal numbers. For the cardinal numbers can be similar to the real numbers, do not specify a general development process, which could all produce. Whether the " totality of all cardinal numbers " a meaningful concept is whether they can be regarded as actual infinity, is controversial even among mathematicians. Interpreted this as a whole lot in terms of axiomatic set theory, namely leads to a logical contradiction (first Cantor's antinomy ).