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Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. In theology, for instance in the work of Duns Scotus, the infinity of God carries the sense not so much of quantity (leading to the question, quantity of what?) but of unconstrainedness. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In popular culture, we find Buzz Lightyear's rallying cry, "To infinity--and beyond!" which might also be called the rallying cry of set theorists considering large cardinals, which are quantitative infinities, defining the number of things in a collection, so large that they cannot be proven to exist in the ordinary mathematics of ZFC, and which might be so large they embody a contradiction.In mathematics, some articles relevant to the subject can be found at limit (mathematics), aleph number, class (set theory), Dedekind infinite, large cardinal, Russell's paradox, hyperreal numbers, projective geometry, extended real number and absolute infinite. In philosophy and theology, one can invesigate the UltimateThe Ultimate is a general term embracing the concept of an ultimate supernatural reality which transcends material reality and from which, according to a broad spectrum of Eastern philosophies and religions, material reality derives. The Ultimate is Gener, the AbsoluteIn some varieties of philosophy, The Absolute describes an ultimate being; the Absolute is the whole of things, all that is. It is usually conceived of as unitary, as spiritual, as conscious — at least insofar as it can be acknowledged by the human mind —, GodThis article focuses on the concept of singular, monotheistic God . See deity, gods, or goddesses for details on divine entities in specific religions and mythologies. God is a term referring to the supreme being generally believed to be ruler or creator, and Zeno's paradoxesZeno's paradoxes are a set of paradoxes conceived by Zeno of Elea to support Parmenides's doctrine that all evidence of the senses is misleading, and particularly that there is no motion. Several of Zeno's eight surviving paradoxes (preserved in Aristotle.
For a discussion about infinity and the physical universe, see UniverseAlternate uses: See Universe (disambiguation In the first half of the 20th century, the word universe was used to mean the whole spacetime continuum in which we exist, together with all the energy and matter within it. Attempts to understand the universe.
1 History
1.1 Ancient view of infinity
The traditional view derives from AristotleAristotle ( Greek Αριστοτλης Aristotelēs) ( 384 BCE March 7, 322 BCE) was a Greek scientist and philosopher. Along with Plato, he is often considered to be one of the two most influential philo:
- "... it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number." [Physics 207b8]
This is often called "potential" infinity, however there are two ideas mixed up with this. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. The other is that we may quantify over finite numbers without restriction. For example "For any integer n, there exists an integer m > n such that P(m)". The second view is found in a clearer form in medieval writers such as William of OckhamWilliam of Ockham (also Occam or any of several other spellings) (ca. 1285- 1349) was a English Franciscan friar and philosopher, from Ockham, a small village in Surrey, near East Horsley. William was devoted to a life of extreme poverty and minimalism.:
- "Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes." (But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent.)
The parts are actually there, in some sense. However, on this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "there are not so many (in number) that there are no more". Aquinas also argued against the idea that infinity could be in any sense complete, or a totality [reference].
