Difference between revisions of "Infinity"

Revision as of 05:15, 31 October 2009

Infinity is an important concept in several disciplines. It connotes some state or entity that is without end. This idea is useful in philosophy, mathematics, the physical sciences and theology.

Mathematics

In mathematics there is no such thing as infinity. The concept of infinity in mathematics is analogous to the concept of aether in physics. Historically it is important; and it permeates the vocabulary of the layman. But it does not correspond to any particular mathematical object or property. Infinity is related to several concepts in mathematics.

Infinity in Set Theory

The most obvious is the idea of an infinite (or non-finite) quantity. A cardinal number is a number corresponding to the size of a set. For example, the set $\{1,2,3\}$ has size 3, or cardinality 3. For finite sets, cardinalities correspond to elements of the set of natural numbers $\mathbb {N} =\{1,2,3,4,\ldots \}$ . However, the set $\mathbb {N}$ itself has more than any finite number of members. We say that $\mathbb {N}$ has infinitely many members, or that it has infinite cardinality. The idea of an infinite cardinality is different than the idea of infinity, because there are infinite sets that have different sizes. For example the set $\mathbb {N}$ has strictly fewer elements than $\mathbb {R}$ , the set of real numbers (as first demonstrated by Georg Cantor). Therefore we need more than one infinite cardinal in order to describe the sizes of sets (in fact we need an infinite number of infinite cardinals). In this way the cardinal numbers can be thought of as an extension of the natural numbers.

We can also extend the natural numbers in a different way to obtain the ordinal numbers. An ordinal number corresponds to an order-isomorphism class of well-ordered sets. For example, again using the set $S=\{1,2,3\}$ we note that we can order S in exactly 6 different ways: $\{1<2<3\},\{1<3<2\},\{2<1<3\},\{2<3<1\},\{3<1<2\},\{3<2<1\}$ . Each of these orderings is essentially the same and indeed any one is order-isomorphic to any other. These orders are part the isomorphism class associated with the ordinal 3 (this class contains all orders on 3 objects). Again, we see that well-orderings of infinite sets do not correspond to any natural number. For instance, we need an ordinal number for the class containing $\mathbb {N}$ , and an ordinal for the class containing $\mathbb {N} \cup \{a\}$ such that $n\leq a$ for all $n\in \mathbb {N}$ , etc. So we have infinitely many infinite ordinals. Note that the principal of mathematical induction on the natural numbers extends to the principal of transfinite induction on the set of ordinal numbers.

Infinity in Calculus

For most of us, our first brush with infinity is our high school calculus course. There infinity is represented by the mathematical symbol $\infty$ . However, calculus is a subset of mathematical analysis, applied to the field $\mathbb {R} ^{n}$ (the Cartesian product of $n$ copies of the real line). Since the real line does not contain an $\infty$ element, infinity does not refer to any specific value or object in calculus. The infinity symbol appears only as a notational convention, and only as part of a "phrase" of the form $x\rightarrow \infty$ (here the infinity symbol denotes that $x$ increases without bound). It also appears as $-\infty$ and $+\infty$ . Again these are simply notational shorthand for "without bound below" and "without bound above" respectively.

Infinity in Topology

Infinity in Algebra/Geometry

Misapplications

Outside mathematics, the concept of infinity is often used by idiots to draw exciting but fallacious conclusions. For example:

The universe is infinite, so everything that can be imagined must exist somewhere! - Er, why?
If you were immortal, you'd eventually have every experience it's possible to have - Or you might just walk round in a big circle the whole time.
Since time is infinite, every prophecy ever made will eventually come true - Great!
To infinity... and beyond!- Buzz Lightyear

Footnotes

@@ Line 1: / Line 1: @@
-'''Infinity''' (or, in mathematical symbols, <math>\mathbf{\infty}</math>) is, in layman terms, the biggest number; well, not really, it's not a number as such - it's bigger than that. The concept of infinity has been one of the most heavily debated philosophical concepts of all time (after the nature of the gods) and is still rejected by some people. Fortunately, modern [[physics]] has helped us out by showing that the [[universe]], as we can measure it, is finite in size, time and the number of particles and such in it - although it does not have to be. This has left infinity to mathematicians and they are liable to get up to anything with it.
+'''Infinity''' is an important concept in several disciplines. It connotes some state or entity that is without end. This idea is useful in philosophy, mathematics, the physical sciences and theology.
-So what is infinity? Well it is easier to define what it is not - it is not finite. The integers from 1 to 7 are finite, there are seven of them. The days of the week are finite, there are also seven of them. The number of fractions between 1 and 7 is not finite. A set <math>S</math> is ''countably'' infinite or ''denumerable'' if, for any <math>n\in\mathbb{N}</math> (''i.e.'', any non-negative integer) there is a corresponding unique member of <math>S</math> - that is, the elements from the set <math>S</math> can be mapped into a one-to-one correspondence with the natural (counting) numbers.  More plainly, if a set is countably infinite, it means we can just make a list of all its elements: for example, the even natural numbers are a countable set, since we can just list them off as "2,4,6,...".  An ''uncountably'' infinite set, like the real numbers <math>\mathbb{R}</math>, is one with too many elements to be put in a such a list.  For example, it turns out that there are so many real numbers that it's impossible to just write them all down one after another: there's no way to talk about the "next" real number after a given one.
+==Mathematics==
-==Now it gets tricky==
+In mathematics there is no such thing as infinity. The concept of infinity in mathematics is analogous to the concept of aether in physics. Historically it is important; and it permeates the vocabulary of the layman. But it does not correspond to any particular mathematical object or property. Infinity is related to several concepts in mathematics.
-Let consider the sequence: 1, 1/2, 1/4, 1/8, and so on.<ref>From now on we will use "..." instead of "and so on".</ref> This sequence is infinite because whenever you find a number in this sequence, such as 1/1024, you can find the next number in the sequence, in this case 1/2048. Lets say that we want to add them all up. Let:
+===Infinity in Set Theory===
-:<math>S=\sum_{i=0}^{\infty}\frac{1}{2^i}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots</math>
-You would expect that adding up an infinite number of numbers should result in infinity, right? However,
-:<math>2S=2+1+\frac{1}{2}+\frac{1}{4}+....</math>
-Then
-:<math>2S-S=2-1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{8}+\dots</math>
-:<math>\therefore S=2</math>
-This property is called convergence. If this does not happen, for example in:
-:<math>R=\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots</math>
-we call the sum divergent, as no matter how many terms we add up, we are not getting closer to a finite number.
-==Counting==
+The most obvious is the idea of an infinite (or non-finite) quantity. A cardinal number is a number corresponding to the size of a set. For example, the set <math>\{1,2,3\}</math> has size 3, or cardinality 3. For finite sets, cardinalities correspond to elements of the set of natural numbers <math>\mathbb{N}=\{1,2,3,4,\ldots\}</math>. However, the set <math>\mathbb{N}</math> itself has more than any finite number of members. We say that <math>\mathbb{N}</math> has infinitely many members, or that it has infinite cardinality. The idea of an infinite cardinality is different than the idea of infinity, because there are infinite sets that have different sizes. For example the set <math>\mathbb{N}</math> has strictly fewer elements than <math>\mathbb{R}</math>, the set of real numbers (as first demonstrated by Georg Cantor). Therefore we need more than one infinite cardinal in order to describe the sizes of sets (in fact we need an infinite number of infinite cardinals). In this way the cardinal numbers can be thought of as an extension of the natural numbers.
-We all should know how to count, but lets look at an example.
-:1 <big>→</big> Sunday
-:2 <big>→</big> Monday
-:3 <big>→</big> Tuesday
-:4 <big>→</big> Wednesday
-:5 <big>→</big> Thursday
-:6 <big>→</big> Friday
-:7 <big>→</big> Saturday
-So there are as many days in the week as there are numbers counting up to 7. Mathematicians call this cardinality, so the days of the week have a cardinality of 7. The arrows represent what mathematicians call a function.<ref>Sorry if this is a bit easy, but this will become obvious in a bit.</ref> So if we can find a function between objects and the number 1,2,3,4..., we can count them.
-It is also important to note that each number corresponds uniquely to one day of the week. Mathematicians call this a ''one-to-one correspondence'' or a ''bijection''.
+We can also extend the natural numbers in a different way to obtain the ordinal numbers. An ordinal number corresponds to an order-isomorphism class of well-ordered sets. For example, again using the set <math>S=\{1,2,3\}</math> we note that we can order S in exactly 6 different ways: <math>\{1<2<3\},\{1<3<2\},\{2<1<3\},\{2<3<1\},\{3<1<2\},\{3<2<1\}</math>. Each of these orderings is essentially the same and indeed any one is order-isomorphic to any other. These orders are part the isomorphism class associated with the ordinal 3 (this class contains all orders on 3 objects). Again, we see that well-orderings of infinite sets do not correspond to any natural number. For instance, we need an ordinal number for the class containing <math>\mathbb{N}</math>, and an ordinal for the class containing <math>\mathbb{N}\cup\{a\}</math> such that <math>n\leq a</math> for all <math>n\in\mathbb{N}</math>, etc. So we have infinitely many infinite ordinals. Note that the principal of mathematical induction on the natural numbers extends to the principal of transfinite induction on the set of ordinal numbers.
-The numbers 1,2,3,4... are called the counting or natural numbers. If we want to show them all, as there are infinitely many, this is written <math>\mathbb{N}</math> and any number is written as just <math>n</math>.
+===Infinity in Calculus===
-Looking back at our earlier sequence we can find a function between <math>\mathbb{N}</math> and the numbers in the sequence:
+For most of us, our first brush with infinity is our high school calculus course. There infinity is represented by the mathematical symbol <math>\infty</math>. However, calculus is a subset of mathematical analysis, applied to the field <math>\mathbb{R}^n</math> (the Cartesian product of <math>n</math> copies of the real line). Since the real line does not contain an <math>\infty</math> element, infinity does not refer to any specific value or object in calculus. The infinity symbol appears only as a notational convention, and only as part of a "phrase" of the form <math>x\rightarrow\infty</math> (here the infinity symbol denotes that <math>x</math> increases without bound). It also appears as <math>-\infty</math> and <math>+\infty</math>. Again these are simply notational shorthand for "without bound below" and "without bound above" respectively.
-:1 <big>→</big> 1
-:2 <big>→</big> 1/2
-:3 <big>→</big> 1/4
-:<math>n</math> <big>→</big> <math>2^{-n}</math>
-We say that the sequence 1,1/2,1/4,... has the same cardinality as <math>\mathbb{N}</math>, since we can count each of them.
+===Infinity in Topology===
-===Now even counting gets tricky===
+===Infinity in Algebra/Geometry===
-Lets consider another infinite set of numbers <math>\textstyle\mathbb{Q}_+</math>, which is every number that looks like <math>\textstyle n/m</math>, with <math>\textstyle n</math> and <math>\textstyle m</math> being in <math>\textstyle\mathbb{N}</math>. When ever <math>\textstyle m=1</math> we have a number that is in <math>\textstyle\mathbb{N}</math> as well as <math>\textstyle\mathbb{Q}_+</math>, so <math>\mathbb{Q}_+</math> contains <math>\textstyle\mathbb{N}</math>, as well as many other numbers, such as 1/2.
-However if we are careful we can find a way of counting <math>\mathbb{Q}_+</math>. By first arranging fractions by their "weight" (n+m) and then size, we can build a function between them.
-:<math>1 \rightarrow 1/1</math>
-:<math>2 \rightarrow 1/2</math>
-:<math>3 \rightarrow 2/1</math>
-:<math>4 \rightarrow 1/3</math>
-:<math>5 \rightarrow 3/1</math>
-:<math>6 \rightarrow 1/4</math>
-:<math>7 \rightarrow 2/3</math>
-:<math>8 \rightarrow 3/2</math>
-:<math>9 \rightarrow 4/1</math>
-So the sets <math>\mathbb{Q}_+</math> and <math>\mathbb{N}</math> are the same "size", even though one is contained with in the other.  The paradox rests in the fact that the concept of size doesn't carry over too well to infinite sets.  A better concept is needed - ''infinite'' cardinalities.
-===Cardinality, the reals, and the continuum hypothesis===
-The cardinality of the natural numbers <math>\mathbb{N}</math> is denoted <math>\aleph_{0}</math> (pronounced "AL-EPH NULL"), and is the lowest cardinality.  That a one-to-one correspondence, a bijection, between the natural numbers and the rational numbers exists means that their cardinalities are the same.  However, no such bijection exists for the real numbers, <math>\mathbb{R}</math>.
-It can be shown that the cardinality of <math>\mathbb{R}</math> is equal to that of the ''power set'' of <math>\mathbb{N}</math>.  The power set is the set of all subsets of any particular set.  Power sets have cardinality <math>2^{S}</math>, where <math>S</math> is the cardinality of the original set.  Thus the reals have cardinality <math>2^{\aleph_{0}}</math>, which is often denoted <math>\mathfrak{c}</math>, the ''cardinality of the continuum''.
-The continuum hypothesis states that there is no cardinal number lying between <math>\aleph_{0}</math> and <math>\mathfrak{c}</math>.  Unfortunately, owing to Godel's incompleteness theorem, this question is unanswerable.  It cannot be proven or disproven.
 ==Misapplications==