INFINITE. — That which has neither end nor limit. If that whose limit is denied is something negative and imperfect, there is a negative i., which coincides with the indeterminate or undefined (potential i.); if, on the other hand, the limit of something positive and perfect is denied, there is a positive i. (actual i.). Potential i. is that which can never be exhausted, whether through increase or division; actual i. is that which contains within itself, at least in a certain order, the fullness of being and determination. Both actual and potential i. may be divided into relative and absolute. Relative i. concerns a particular genus of being, such as mathematical i. and, in Thomistic doctrine, the angelic essence (Sum. Theol., q. 50, a. 2, ad 4). Absolute i., on the other hand, refers to all being without restriction: prime matter, pure potency and indeterminacy, as potential i., and God, pure act, perfection and absolute fullness, as actual i.
The notion of i. is fundamental in philosophy and science, from the time when Anaximander made i. (ἄπειρον) the principle from which everything derives; for, as Aristotle observes (Phys., III, 4, 205 a 1-b 15), all the best philosophers of antiquity spoke of i. as the principle of beings: from the Ionian physicists, with the infinity of the single original matter, to Melissus, who extends the immobile sphere of Parmenides to i.; to Anaxagoras and the atomists, who posit an infinite number of elements in an infinite space; from the Pythagoreans, with the obscure doctrine of even number as the root of i., to Plato, who transforms the Pythagorean i. into the indeterminate dyad of the great and the small. But the antinomies of the finite and the i. soon appeared to the Greek mind, as is evident from Zeno’s famous arguments against motion and divisibility (Aristotle, Phys., VI, 9, 239 b 5-40 a 18). The solution of this antinomy led Aristotle to the clear distinction between actual and potential i. and to the denial of every actual i. in matter (ibid., III, 5-7, especially 206 a 9-25); nevertheless, he also recognized, at least incidentally, positive actual infinity in the First Mover, i. in time and in power (ibid., VIII, 10, 267 b 17-26), the best of all beings (Met., XII, 7, 1072 b 24-30). The deepening of the notion of actual i. in all its positivity was, however, the work of Christian philosophy, which recognizes infinity as one of the divine attributes, according to the dogmatic constitution of the First Vatican Council (Denz-U, 1782).
In modern philosophy the problem of the genesis and real value of the concept of i. was explicitly posed. For while sensationalism and empiricism (Locke, Hobbes) simply denied the notion of actual and positive i., ontologism, on the contrary, demanded for the human intellect the immediate intuition of Infinite Being, since the finite cannot represent i., indeed cannot even be conceived as such, unless the intuition of i. precedes II. According to traditional philosophy, on the other hand, the human intellect has the concept of i., but only indirectly, analogically, and through a process of double negation, insofar as, from consideration of the various degrees of reality, it retains not only the positive concept of entity and perfection, but also the negative concept of distinction and consequently of limit. With the negation of this limit the concept of i. arises. Yet, in the absence of direct intuition, it is not possible to pass from the idea of i. to the affirmation of its reality, or even of its possibility; herein lies the internal weakness of the Anselmian argument, which infers God’s real existence from the concept of divine infinity.
In recent times the development of mathematics has once again brought to the fore the antinomies of the infinitely great and the infinitely small, first through infinitesimal calculus and then through Cantor’s transfinites. No less lively at all times has been the antinomy between the finite and i. in physical space, which returned to discussion on the basis of the Copernican system, Newtonian mechanics, the more recent results of astronomy, and recent physical theories, particularly the theory of relativity, which posits space as finite but not limited.
These are the principal historical developments and problems concerning the notion of i.; for their treatment V. ATTRIBUTI DIVINI; CONTINUO; LIMITE; MOTO; RELATIVITÀ; UNIVERSO.