Математика философиясындағы нақты шексіздік тұжырымы

Concept in the philosophy of mathematics

In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. As a result, potential infinity is often formalized using the concept of a limit.

The ancient Greek term for the potential or improper infinite was apeiron (unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon. Apeiron stands opposed to that which has a peras (limit). These notions are today denoted by potentially infinite and actually infinite, respectively. Anaximander (610–546 BC) held that the apeiron was the principle or main element composing all things. Clearly, the 'apeiron' was some sort of basic substance. Plato's notion of the apeiron is more abstract, having to do with indefinite variability. The main dialogues where Plato discusses the 'apeiron' are the late dialogues Parmenides and the Philebus.

Aristotle handled the topic of infinity in Physics and in Metaphysics. He distinguished between actual and potential infinity. Actual infinity is completed and definite, and consists of infinitely many elements. Potential infinity is never complete: elements can be always added, but never infinitely many. "For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different." |Aristotle, Physics, book 3, chapter 6. Aristotle distinguished between infinity with respect to addition and division. But Plato has two infinities, the Great and the Small. |Physics, book 3, chapter 4. "As an example of a potentially infinite series in respect to increase, one number can always be added after another in the series that starts 1,2,3, but the process of adding more and more numbers cannot be exhausted or completed. "With respect to division, a potentially infinite sequence of divisions might start, for example, 1, 1/2, 1/4, 1/8, 1/16, but the process of division cannot be exhausted or completed. "For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately." | Metaphysics, book 9, chapter 6. Aristotle also argued that Greek mathematicians knew the difference among the actual infinite and a potential one, but they "do not need the [actual] infinite and do not use it" (Phys. III 2079 29).

Actual infinity is now commonly accepted, because mathematicians have learned how to construct algebraic statements using it. For example, one may write down a symbol, , with the verbal description that " stands for completed (countable) infinity". This symbol may be added as an ur element to any set. One may also provide axioms that define addition, multiplication and inequality; specifically, ordinal arithmetic, such that expressions like can be interpreted as "any natural number is less than completed infinity". Even "common sense" statements such as are possible and consistent. The theory is sufficiently well developed, that rather complex algebraic expressions, such as , and even can be interpreted as valid algebraic expressions, can be given a verbal description, and can be used in a wide variety of theorems and claims in a consistent and meaningful fashion. The ability to define ordinal numbers in a consistent, meaningful way, renders much of the debate moot; whatever personal opinion one may hold about infinity or constructability, the existence of a rich theory for working with infinities using the tools of algebra and logic is clearly in hand.

The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential, but not to be mistaken for physically existing. The question of whether natural or real numbers form definite sets is therefore independent of the question of whether infinite things exist physically in nature. Proponents of intuitionism, from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets. Consequently, they reconstruct the foundations of mathematics in a way that does not assume the existence of actual infinities. On the other hand, constructive analysis does accept the existence of the completed infinity of the integers. For intuitionists, infinity is described as potential; terms synonymous with this notion are becoming or constructive. To access memory on the tape, a Turing machine moves a read head along it in finitely many steps: the tape is therefore only "potentially" infinite, since — while there is always the ability to take another step — infinity itself is never actually reached. Mathematicians generally accept actual infinities. Georg Cantor is the most significant mathematician who defended actual infinities. He decided that it is possible for natural and real numbers to be definite sets, and that if one rejects the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then one is not involved in any contradiction. The present day conventional finitist interpretation of ordinal and cardinal numbers is that they consist of a collection of special symbols, and an associated formal language, within which statements may be made. All such statements are necessarily finite in length. The soundness of the manipulations is founded only on the basic principles of a formal language: term algebras, term rewriting, and so on. More abstractly, both (finite) model theory and proof theory offer the needed tools to work with infinities. One does not have to "believe" in infinity in order to write down algebraically valid expressions employing symbols for infinity.

The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound. Classical set theory accepts the notion of actual, completed infinities. However, some finitist philosophers of mathematics and constructivists object to the notion. If the positive number n becomes infinitely great, the expression 1/n goes to naught (or gets infinitely small). In this sense one speaks of the improper or potential infinite. In sharp and clear contrast the set just considered is a readily finished, locked infinite set, fixed in itself, containing infinitely many exactly defined elements (the natural numbers) none more and none less. (A. Fraenkel [4, p. 6])Thus the conquest of actual infinity may be considered an expansion of our scientific horizon no less revolutionary than the Copernican system or than the theory of relativity, or even of quantum and nuclear physics. (A. Fraenkel [4, p. 245])

To look at the universe of all sets not as a fixed entity but as an entity capable of "growing", i. e., we are able to "produce" bigger and bigger sets. (A. Fraenkel et al. [5, p. 118])

(Brouwer) maintains that a veritable continuum which is not denumerable can be obtained as a medium of free development; that is to say, besides the points which exist (are ready) on account of their definition by laws, such as e, pi, etc. other points of the continuum are not ready but develop as so called choice sequences. (A. Fraenkel et al. [5, p. 255])

Intuitionists reject the very notion of an arbitrary sequence of integers, as denoting something finished and definite as illegitimate. Such a sequence is considered to be a growing object only and not a finished one. (A. Fraenkel et al. [5, p. 236])

Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for "actual infinity." The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets. (T. Jech )

Owing to the gigantic simultaneous efforts of Frege, Dedekind and Cantor, the infinite was set on a throne and revelled in its total triumph. In its daring flight the infinite reached dizzying heights of success. (D. Hilbert [6, p. 169])

One of the most vigorous and fruitful branches of mathematics [ ] a paradise created by Cantor from which nobody shall ever expel us [ ] the most admirable blossom of the mathematical mind and altogether one of the outstanding achievements of man's purely intellectual activity. (D. Hilbert on set theory [6])

Finally, let us return to our original topic, and let us draw the conclusion from all our reflections on the infinite. The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking – a remarkable harmony between being and thinking. (D. Hilbert [6, 190])

Infinite totalities do not exist in any sense of the word (i. e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. (A. Robinson [10, p. 507])

Indeed, I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world. (A. Robinson)

Georg Cantor's grand meta narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory. (Y. Manin )

Thus, exquisite minimalism of expressive means is used by Cantor to achieve a sublime goal: understanding infinity, or rather infinity of infinities. (Y. Manin )

There is no actual infinity, that the Cantorians have forgotten and have been trapped by contradictions. (H. Poincaré [Les mathématiques et la logique III, Rev. métaphys. morale 14 (1906) p. 316])

When the objects of discussion are linguistic entities [ ] then that collection of entities may vary as a result of discussion about them. A consequence of this is that the "natural numbers" of today are not the same as the "natural numbers" of yesterday. (D. Isles )

There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity regarding the numbers as an incomplete infinity offers a viable and interesting alternative to regarding the numbers as a completed infinity, one that leads to great simplifications in some areas of mathematics and that has strong connections with problems of computational complexity. (E. Nelson )

During the renaissance, particularly with Bruno, actual infinity transfers from God to the world. The finite world models of contemporary science clearly show how this power of the idea of actual infinity has ceased with classical (modern) physics. Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing. Within the intellectual overall picture of our century actual infinity brings about an impression of anachronism. (P. Lorenzen)