The Infinity reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

# Infinity

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Infinity (from the Latin "infinitus", meaning unlimited, and usually denoted by the symbol ∞) is the quality of being unbounded or having no limit. The infinite is usually defined as that which has no bounds in space or time.

## Ancient view of infinity

"... 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 Phi(m)". The second view is found in a clearer form in medieval writers such as William of Ockham:

"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].

## Early modern views

Galileo (during his long house arrest in Sienna after his condemnation by the Inquisition) was the first to notice that we can place a set of infinite numbers into one-to-one correspondence with one of its proper subsets (any part of the set, that is not equivalent to the whole). For example, we can match up the "set" of even numbers {2, 4, 6, 8 ...} with the natural numbers {1, 2, 3, 4 ...} as follows

1, 2, 3, 4, ...
2, 4, 6, 8, ...

It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same size. He thought this was one of the difficulties which arise when we try, "with our finite minds", to comprehend the infinite.

"So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal", "greater", and "less", are not applicable to infinite, but only to finite, quantities." [On two New Sciences, 1638]

The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets.

Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. They believed all our ideas were derived from sense appearance or "impressions", and since all sense impression is inherently finite, so too for our thoughts and ideas. Our idea of infinity is merely negative or privative.

"Whatever positive ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity ... yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused , because it is made up of two parts very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression." (Essay, II. xvii. 7., author's emphasis)

Famously, the ultra-empiricist Hobbes tried to defend the idea of a potential infinity in the light of the discovery by Evangelista Torricelli, of a figure (Gabriel's horn) whose surface area is infinite, but whose volume is finite.

## Mathematical conception

In mathematics, a distinction is made between different "grades" of infinity because it can be shown that some infinite sets have greater cardinality than others. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (), the cardinality of the set of natural numbers.

The modern mathematical conception of the infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of sets. Their approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its "proper" parts.

Thus Cantor showed that infinite sets can even have different sizes, distinguished between countably infinite and uncountable sets, and developed a theory of cardinal numbers around this. His view prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the surreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Our intuition gained from finite sets breaks down when dealing with infinite sets. One example of this is Hilbert's paradox of the Grand Hotel.

It is worth mention that the infinite cardinal numbers (relating to set theory) and the infinity commonly encountered in algebra and calculus are two completely different concepts. In algebra and calculus, 2 is not technically a number, but taking a limit yields 2 = ∞. Real numbers are not used to measure the sizes of sets, so ∞ can be used for any quantity that grows indefinitely at a limit. The corresponding statement in set theory is that 20 > ℵ0 because the former term is uncountable, while the latter is countable. Exponents in set theory are not the same as regular exponents in high school mathematics, and ∞ is not the same as aleph null.

An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might have a similar topology; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point.

## Modern philosophical views

Modern discussion of the infinite is now regarded as part of set theory and mathematics, and generally avoided by philosophers. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period". (see also Logic of antinomies )

"Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes ... In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar." (Philosophical Remarks ç 141, cf Philosophical Grammar p. 465)

Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience.

"... I can see in space the possibility of any finite experience ... we recognise [the] essential infinity of space in its smallest part." "[Time] is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room."

"... what is infinite about endlessness is only the endlessness itself."

## Mathematics without infinity

Leopold Kronecker rejected the notion of infinity and started a branch of mathematics, called finitism (see also mathematical constructivism).

## The Absolute

Another question is whether the mathematical conception of infinity has any relation to the religious concept of God. This question was addressed by both Cantor, with his concept of the Absolute Infinite which he equated with God, and Kurt Gödel with his "ontological proof" of the existence of an entity he related to God.