Infinity

From RationalWiki

Infinity (or, in mathematical symbols, $\mathbf{\infty}$ ) 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.

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 $S$ is countably infinite or denumerable if, for any $n\in\mathbb{N}$ (i.e., any non-negative integer) there is a corresponding unique member of $S$ - that is, the elements from the set $S$ 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 $\mathbb{R}$ , 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.

[edit] Now it gets tricky

Let consider the sequence: 1, 1/2, 1/4, 1/8, and so on.^[1] 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:

$S=\sum_{i=0}^{\infty}\frac{1}{2^i}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$

You would expect that adding up an infinite number of numbers should result in infinity, right? However,

$2S=2+1+\frac{1}{2}+\frac{1}{4}+....$

Then

$2S-S=2-1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{8}+\dots$

$\therefore S=2$

This property is called convergence. If this does not happen, for example in:

$R=\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots$

we call the sum divergent, as no matter how many terms we add up, we are not getting closer to a finite number.

[edit] Counting

We all should know how to count, but lets look at an example.

1 → Sunday

2 → Monday

3 → Tuesday

4 → Wednesday

5 → Thursday

6 → Friday

7 → 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.^[2] 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.

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 $\mathbb{N}$ and any number is written as just $n$ .

Looking back at our earlier sequence we can find a function between $\mathbb{N}$ and the numbers in the sequence:

1 → 1

2 → 1/2

3 → 1/4

n

→

2 - n

We say that the sequence 1,1/2,1/4,... has the same cardinality as $\mathbb{N}$ , since we can count each of them.

[edit] Now even counting gets tricky

Lets consider another infinite set of numbers $\textstyle\mathbb{Q}_+$ , which is every number that looks like $\textstyle n/m$ , with $\textstyle n$ and $\textstyle m$ being in $\textstyle\mathbb{N}$ . When ever $\textstyle m=1$ we have a number that is in $\textstyle\mathbb{N}$ as well as $\textstyle\mathbb{Q}_+$ , so $\mathbb{Q}_+$ contains $\textstyle\mathbb{N}$ , as well as many other numbers, such as 1/2.

However if we are careful we can find a way of counting $\mathbb{Q}_+$ . By first arranging fractions by their "weight" (n+m) and then size, we can build a function between them.

$1 \rightarrow 1/1$

$2 \rightarrow 1/2$

$3 \rightarrow 2/1$

$4 \rightarrow 1/3$

$5 \rightarrow 3/1$

$6 \rightarrow 1/4$

$7 \rightarrow 2/3$

$8 \rightarrow 3/2$

$9 \rightarrow 4/1$

So the sets $\mathbb{Q}_+$ and $\mathbb{N}$ 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.

[edit] Cardinality, the reals, and the continuum hypothesis

The cardinality of the natural numbers $\mathbb{N}$ is denoted $\aleph_{0}$ (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, $\mathbb{R}$ .

It can be shown that the cardinality of $\mathbb{R}$ is equal to that of the power set of $\mathbb{N}$ . The power set is the set of all subsets of any particular set. Power sets have cardinality $2 S$ , where $S$ is the cardinality of the original set. Thus the reals have cardinality $2^{\aleph_{0}}$ , which is often denoted $\mathfrak{c}$ , the cardinality of the continuum.

The continuum hypothesis states that there is no cardinal number lying between $\aleph_{0}$ and $\mathfrak{c}$ , that is to say, there's no set with more elements than the integers but fewer elements than the real numbers. This turns out to be a very tricky claim, and now it's known that although the continuum hypothesis can't be proved using the usual axioms of mathematics (i.e. ZFC), it also won't create any contradictions if we assume it.

[edit] 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

[edit] Footnotes

↑ From now on we will use "..." instead of "and so on".
↑ Sorry if this is a bit easy, but this will become obvious in a bit.

[0] From now on we will use "..." instead of "and so on".

[1] Sorry if this is a bit easy, but this will become obvious in a bit.

[1]

[2]

Infinity

From RationalWiki

Contents

[edit] Now it gets tricky

[edit] Counting

[edit] Now even counting gets tricky

[edit] Cardinality, the reals, and the continuum hypothesis

[edit] Misapplications

[edit] Footnotes

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