Infinity

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\frac{d}{dx} \sin x=?\, This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

Infinity may be defined as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount."[1]

That is, it is something which "goes on without end". Infinity is denoted by this symbol:\infty\,, looking like an 8 on its side.

Infinity comes up in many aspects of mathematical discourse and is often treated as a number, but it is not a real number. In mathematical contexts, infinity often appears as a limit, as an integral, as the measure (size) of a set in Euclidean space, and as the cardinality (size) of sets in general.

The simplest place to see this seeming paradox is in the fact that all integers are finite, but that there are an infinite number of them.

  • Why are all integers finite? Because, if infinity were an integer, what would \infty+1\, be?
  • Why is the set of all integers an infinite set? Because they go on forever. If the set were finite, there would be a biggest integer. But that can't be, because we can always add one to any integer. This kind of thinking is often schoolchildren's first introduction to logical reasoning.

While infinity is not a number, it appears in many other contexts. As we have seen, the size of the set of integers is infinite. We say that the cardinality of the integers (this set is often denoted \mathbb{Z}) is infinite. The rational numbers (\mathbb{Q}) and the real numbers (\mathbb{R}) also have infinite cardinality. An interesting result from set theory (see below) is that \mathbb{Z} and \mathbb{Q} have the same cardinality, while \mathbb{R} has larger cardinality than the other two.

Another place where infinity arises is in limits. We can say that "the limit of 1/x\, as x\, approaches zero is infinity", or "the limit of e^{-x}\, as x\, approaches infinity is zero", but this is because infinity has a special meaning in the context of limits. See limit for discussion of this.

We can also say that "the measure (size) of the set of reals is infinity", or that certain integrals are infinite, but this is because of special properties of measures and integrals.

So infinity might arise in statements like these:

  • \frac{1}{0} = \infty\ \ NO! This isn't allowed! Infinity is not a number, and division by zero is illegal!
  • \lim_{x\to 0}\frac{1}{x} = \infty\ \ Yes, sort of. One could say that "the limit is infinite", since that includes both positive and negative infinite values. But to say that the limit is "equal to infinity", one would have to say that this is the limit from the right. The limit from the left is "minus infinity", that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above.
  • \int_0^1\frac{1}{x} = \infty\ \ Infinity has a special meaning for integrals.
  • \|\mathbb{Z}\| = \infty\ \ The cardinality of the integers is infinite.
  • \mu(\mathbb{R}) = \infty\ \ The (Lebesgue) measure of the reals is infinite.

In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible. It can't be done in general.

Another place where infinity is considered acceptable is in Floating-point arithmetic in computers. The IEEE-754 standard (which all modern computers support) allows the values \infty\, and - \infty\,. But their treatment is very different from other numbers, and computer floating-point numbers don't faithfully model the real numbers in any case.

Also, in some non-standard models of Peano Arithmetic, \infty\, is treated as an actual number. But, once again, this is not standard mathematics.

Georg Cantor's diagonal argument is an elegant proof demonstrating that the (infinite) cardinality of real numbers is greater than the (infinite) cardinality of countable integers. The essence of the argument is that in any proposed list of all real numbers, a new real number not in the list can be constructed by taking the digits in a diagonal through the list and changing them to construct a new real number that differs from the nth entry at the nth position right of the decimal point.

To formalize the concepts of countably and uncountably infinite sets, we need the set theory concept of cardinality. Using this concept it can be shown that there are infinitely many distinct infinite cardinalities.

In Zermelo-Fraenkel set theory, there is the Axiom of Infinity, asserting the existence of an infinite set. The set that it creates is essentially the same as the integers. Some "constructive" mathematicians work without this axiom, and determine which results may be proved without assuming it.

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Classification of infinities

In modern mathematics, it is recognized that there is no single concept of 'infinity' - there are many different infinities. We can divide these into some broad classes. Firstly, there are the cardinals, which are used to measure the sizes of sets; and then there are ordinals, which are used to measure the position of an item in an ordered list. For finite quantities, we can use the same numbers for both - a race can have 3 participants, and you can come 3rd in the race. But, with infinite quantities, this no longer applies, the same numbers can now be used for both purposes. Thus, the smallest infinite cardinal is aleph-null, but the smallest infinite ordinal is omega.

Cardinals

We say two sets have the same size if it is possible to put their respective members into one-to-one correspondence. A cardinal can be conceived as the equivalence class of all sets having that size.

Consider the set of all natural numbers, and its subset the set of all odd natural numbers. One would think that there are half as many odd numbers as natural numbers. Yet, while that is true when dealing with some finite subset of the natural numbers, when dealing with their entirety, it turns out that there are as many odd numbers as natural numbers. This can be shown by the fact that the natural numbers and the odd natural numbers can be put in a one-to-one correspondence: 0 -> 1, 1 -> 3, 2 -> 5, 3 -> 7, etc. (represented by the formula 2*n + 1). Thus, whereas for finite sets if A is a proper subset of B, then the cardinality of A must be less than the cardinality of B, for infinite sets this is no longer the case; an infinite set may be the same size as one of its proper subsets.

The cardinality of the natural numbers is known as aleph-null. There are as many integers as natural numbers; and as many rational numbers as natural numbers. But it can be shown, using Cantor's diagonalization argument, that the reals are strictly larger than the natural numbers - you cannot put the natural numbers into one to one correspondence with the reals. The cardinality of the reals is the cardinality of the power set of the natural numbers, which is known as the power of the continuum (c) or as beth-one. (Aleph-null is the same as beth-null; whether aleph-one is the same as beth-one is a mathematical question, which essentially has no answer - and we can even show it has no answer.)

The generalised continuum hypothesis (GCH) states that \beth_{a} = \aleph_{a} for all ordinals a. So, the continuum hypothesis is a special case of the generalised continuum hypothesis for ordinal 0. Like CH, GCH is independent of the axioms of ZFC.

A set having cardinality equal to or less than \aleph_{0} is called countable; a set of greater cardinality is called uncountable.

Ordinals

Numbers serve two distinct purposes - to measure the size of sets, and to measure the position of an item in an ordering. Thus, we may speak of a race having 2, 3, 4, 5, etc. contestants, and we can speak of the contestants as having come 1st, 2nd, 3rd etc. (or even 0th, if one is a mathematician!) When we measure the number of elements in a set, we are using cardinal numbers; when we measure an item's position in an ordering, we are using ordinal numbers. For finite quantities, it does not make that much difference, since for finite quantities we can use the same numbers to serve both purposes. But for transfinite quantities, that is no longer the case - we can no longer use the same numbers as both cardinals and ordinals. Thus, the smallest transfinite cardinal is \aleph_{0}, but the smallest transfinite ordinal is ω.

Other types of infinite numbers

Other than transfinite cardinals and ordinals, there are other types of infinite numbers:

  • the affinely extended reals: these are elements of the set \mathbb{R} \cup \{ +\infty, -\infty \}. In other words, the real numbers extended by the addition of positive and negative infinity. +\infty is greater than every real number, and -\infty is less than every real number.
  • the projectively extended reals: these are elements of the set \mathbb{R} \cup \{ \infty \}. In other words, the real numbers extended by the addition of a single signless infinity. The ordering relation is not defined for the projectively extended reals - but informally, \infty can be said to be both greater than and less than every real number.
  • the hyperreals: denoted *\mathbb{R}, these extend the real numbers with both infinite and infinitesimal quantities. They are used in non-standard analysis. Other extensions of the real numbers with infinite and infinitesimal quantities include the surreal numbers, the superreal numbers, and the Levi-Civita field.

The conclusion to be drawn from all of this, is that although in popular usage we speak about infinity as if it were a single thing, there are actually numerous diferent infinities, all belonging to different systems. If we want to talk about infinity, we ought to be careful to specify which infinity we mean.

Archimedean property

An ordered field is said to be Archimedean if it has no infinitely large or infinitely small elements. One way of stating this: an ordered field is Archimedian, if for every element of the field, there is a greater natural number. This is true for the reals - for every real number, there is a greater natural number - but not for example for the affinely extended reals, since +\infty is greater than every natural number. So the affinely extended reals are said to be non-Archimedian. Likewise, the hyperreal numbers, the surreal numbers, the superreal numbers and the Levi-Civita field are all non-Archimedian. Any ordered field containing infinities must also contain infinitesimals as their inverse.

To understand the notion of non-Archimedeanity, it is interesting to imagine what it would be like were space or time were non-Archimedean. If space was non-Archimedean, there could be areas of the universe infinitely distant from here. If time was non-Archimedean, there could be past times an infinite number of years before the present, and future times an infinite number of years into the future. It appears that no one has every seriously suggested that time or space actually are non-Archimedean, but it is nonetheless an interesting mental exercise to understand the notion.

Philosophy of mathematics

Different positions in the philosophy of mathematics have different attitudes towards infinity. At one extreme, Platonists and formalists generally have no philosophical to any form of infinity than can be defined. At the other extreme, finitists and ultrafinitists deny the existence of infinity; to them, only finite quantities, and finite objects, actually exist. In the middle, many constructivists and intuitionists adopt a position of countablism - an acceptance of the existence of countable sets, but denying that uncountable sets exist.

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