Actual Infinity - Classical Set Theory

Classical Set Theory

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 )

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 )

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. )

(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. )

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. )

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 )

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 )

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 )

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 )

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 )

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. (E. Nelson )

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)