It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'." If you can't find the critical points then just check from both end point. Find the image of the triangle having vertices(1, 2),(3, 4),(4, 6)under the translation that takes the point(1, 2) to(9, 1) Or is it an imaginary number. All absolute value tells you is the distance away from 0 something is on the number line (so negatives become positive and positives stay positive). The absolute value of infinity will just be positive infinity. This means there is only a (developing, improper, "syncategorematic") potential infinity but not a (fixed, proper, "categorematic") actual infinity. More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. Gavin says: July 18, 2017 at 3:27 pm. "It is well known that in the Middle Ages all scholastic philosophers advocate Aristotle's "infinitum actu non datur" as an irrefutable principle." (G.W. Note that absolute infinity is a well-ordered class by itself if we identify it with \(\textrm{Ord}\). One positive integer is 7 less than twice another. (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. The ancient Greek term for the potential or improper infinite was apeiron (unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon. (Georg Cantor)[10] (G. Cantor [8, p. 252]), One proof is based on the notion of God. [9] (C.F. 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. Within the intellectual overall picture of our century ... actual infinity brings about an impression of anachronism. Yes, absolute infinity times 2 Seriously, there is no such thing as absolute infinity. Aristotle handled the topic of infinity in Physics and in Metaphysics. In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. (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. [11] For example, Stephen Kleene describes the notion of a Turing machine tape as "a linear 'tape', (potentially) infinite in both directions. (Aristotle). 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. Is there anything greater than absolute infinity? (Y. Manin [3]), There is no actual infinity, that the Cantorians have forgotten and have been trapped by contradictions. For instance, f(x)=1/(1+x 2) is a continuous function defined for all real numbers x, and it also tends to a limit of 0 when x "goes to" plus or minus infinity (in the sense of potential infinity, described earlier). 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. For example, it is used in the transfinite induction and in the definition of Little Bigeddon. (J. Baconthorpe [9, p. 96]). What is the absolute value of infinity? From the nature of time – for it is infinite. There were exceptions, however, for example in England. (R. Dedekind [3a, p. III]). The majority[citation needed] agreed with the well-known quote of Gauss: I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Still have questions? These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. (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 [6, 190]), Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). Stephen Kleene 1952 (1971 edition):48 attributes the first sentence of this quote to (Werke VIII p. 216). No matter how lightweight its material, an, Actual infinity follows from, for example, the acceptance of the notion of the integers as a set, see J J O'Connor and E F Robertson, [, Learn how and when to remove this template message, "The Definitive Glossary of Higher Mathematical Jargon — Infinite", "Logos Virtual Library: Aristotle: Physics, III, 7", "Infinity" at The MacTutor History of Mathematics archive,, Cleanup tagged articles with a reason field from June 2010, Wikipedia pages needing cleanup from June 2010, Articles with unsourced statements from November 2019, Articles with unsourced statements from August 2009, Creative Commons Attribution-ShareAlike License.