Different kinds of Mathematical infinities

For talk about all things science.

Different kinds of Mathematical infinities

Postby jjb2011 » Sun Sep 17, 2017 1:15 am

It was great to read a story in the Times and a leading article on Friday 15th September concerning the nature of mathematical infinities. It turns out that some infinities are bigger than others, and some infinities, although they look like they are different in size, turn out to be the same size.
A good example of an infinite set is the set of all natural numbers, 1,2,3,4,5........and so on. Clearly this set is infinite. But what about the set of even numbers, 2,4,6,8.......and so on. Clearly this set of numbers is infinite as well, but this set, at first sight,has only half the numbers as the set of natural numbers. So, is the infinity of natural numbers twice the size of the infinity of even numbers ?
It turns out that these infinities are the same size. The explanation is due to Cantor. For these two sets of numbers, the numbers can be paired up. so for example .....
1 --> 2
2 --> 4
3 --> 6
4 --> 8 and so on.
So for every number in the set of natural numbers there is a corresponding number in the set of even numbers. Because of the this correspondence of elements the two sets which are both infinite, have the same size. These infinities are countable and one countable infinite set is the same as any other countable infinite set.

The situation is different when we look at real numbers. Real numbers are decimal numbers ..... 0.537892562 for example. The interesting thing about real numbers is that if you take any two real numbers, there are an infinite number of real numbers between them.It doesn't matter how close you choose the initial two real numbers. So the infinite set of real numbers is infinitly bigger than the infinite set of natural numbers. The real numbers cannot be put into a list and are not countable.


Hi, you are viewing as a guest, You can only see this first post in this topic. if you sign up you get access to other goodies you can't even see as a guest, including video tutorials on imaging and processing, scope modifications and even member discounts on gear!

So, give us a try, what have you got to lose! Oh, and if you stay, when you reach 50 posts you get access to MORE goodies! What are you waiting for!

Black Dwarf
Posts: 1213
Joined: Tue Jun 21, 2011 10:21 pm
Location: Ledbury, Herefordshire

Re: Different kinds of Mathematical infinities

Postby galileo » Tue Oct 24, 2017 7:33 pm

User avatar
Black Dwarf
Posts: 4249
Joined: Mon Jan 28, 2013 6:55 pm
Location: Chorlton cum Hardy Manchester

Re: Different kinds of Mathematical infinities

Postby carastro » Tue Oct 24, 2017 9:09 pm

User avatar
Black Dwarf
Posts: 5207
Joined: Sun Jul 03, 2011 9:24 pm
Location: Bromley Kent

Re: Different kinds of Mathematical infinities

Postby pixelsaurus » Mon Dec 04, 2017 8:16 pm

User avatar
Red Giant
Posts: 371
Joined: Sun Jul 08, 2012 2:12 pm
Location: New Zealand

Re: Different kinds of Mathematical infinities

Postby Mrs Mangel » Sun Jan 14, 2018 10:34 pm

Mrs Mangel
Posts: 85
Joined: Sun Feb 01, 2015 9:38 pm


Who is online

Users browsing this forum: No registered users and 2 guests