Math Is Fun Forum

_-UnEnLighTenEd One-_

Guest

different sizes of infinity

hi,

a friend told me that you can have different sizes of infinity. i don't believe him, i mean, if you have infinity divided by infinity, you can't get a definite answer becuase not only are neither real numbers, but to have an infinitely large constant on the top and bottom dont make sense.

so how can you have different sizes of infinity help please.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: different sizes of infinity

Hi;

Cantor proved a while back that not all infinities are the same size. Some infinite sets are bigger than others.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: different sizes of infinity

Consider the sequence (1, 2, 3, 4, 5, ...), and compare it with (2, 4, 6, 8, 10, ...).

Both of those go to infinity, but if you divide the two nth terms then you'll always get 2. Tweak the second sequence and you can get infinity/infinity to equal whatever you want (except 0).

However, if you also use (1, 4, 9, 16, 25, ...) then you can get an answer of 0, and also of infinity.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: different sizes of infinity

Before you start considering sequences, it is best to say what we mean by the "size of infinity".  Of course when we say this, we mean the size of an infinite set.

So we can begin with something you know: the size of a finite set.  For example: {a, b, c} and {1, 2, 3}.  These sets have the same size.  You could say that their size is three, but the genius Cantor discovered that you don't need numbers to talk about the relative sizes of these sets.  We say that two sets are the same size if you can pair elements together so that:

(a) You don't use a single element twice and (b) every element is used.

For example, a pairing from the sets above is: {a, 1}, {b, 2}, {c, 3}

It should be obvious that if two finite sets have the same size, then you can pair their elements in this way.  And if course, if two finite sets can be paired, then they have the same size.

Now here is the kicker: When it comes to infinite sets, we can't describe their sizes by a number.  But we can still try to pair elements.  For mathsyperson's example, we have the natural numbers and the even natural numbers.  Here is a pairing:

{1, 2}, {2, 4}, {3, 6}, {4, 8}, {5, 10}, ..., {n, 2n}, ...

I use every element from each set only once, and I used every single element from each set.  Therefore, the natural numbers have the same size as the even natural numbers.  Now this may seem odd, because the even numbers are properly contained inside the natural numbers, but infinity is odd, so that's ok.

Cantor proved that you can't pair the integers with the real numbers.  No such pairing can exist.  This is known as Cantor's Diagonal Argument.  This is why we say the real numbers are bigger than the integers, there is absolutely no way to pair them.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: different sizes of infinity

Hi _-UnEnLighTenEd One-_;

See if you can find on the internet or elsewhere anything by Raymond Smullyan. He provides a good beginners understanding of denumerability using his charming girl and hell example. It is not as precise as Cantor's proof and it can be cumbersome, but it is a start.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.