Beyond Infinity? (Part II)

This is Part II of a popular maths talk by Dr Joel Feinstein, University of Nottingham (Blog: http://explainingmaths.wordpress.com/ ).
This part is more technical than the first, and it is probably best suited to first-year mathematics undergraduates, or advanced maths A level students. Others may find the technical details hard to follow, and should focus on the overview.
The full talk gives an introduction to various different kinds of infinity, both countable and uncountable. These concepts are illustrated in a somewhat informal way using the notion of Hilbert's infinite hotel. In this talk, the hotel manager tries to fit various infinite collections of guests into the hotel. From the talk, you should learn that many apparently different types of infinity are really the same size. However, there are genuinely "more" real numbers than there are positive integers, as is shown in this more challenging final section (Part II), using Cantor's diagonalization argument. See http://unow.nottingham.ac.uk/ for more resources from Nottingham. See http://www.maths.nottingham.ac.uk/personal/jff/Beamer/ for more of Joel Feinstein's resources, including a higher quality video of the full version of this talk.

Category:

Education

Tags:

• Hilbert
• Grand
• Hotel
• popular
• maths
• math
• countable
• uncountable

[JoelFeinstein]

A newer edition of this video is available

(see the March 2010 edition)

[anticorncob6]

You know how all the real numbers can be squeezed into the real numbers between 0 and 1?

Squeeze in the numbers from 0 - 1 between 0 and 1/2. Between 1/2 and 3/4, you squeeze in 0 to -1. 0 to 2 go between 3/4 and 7/8, and so on.

So, how could you get all the (x,y) values into the continuum hotel when x and y are real numbers? I proved it is possible but I do not know a way.

[JoelFeinstein]

@anticorncob6

The usual method is to alternate between their decimal expansions

So for example the pair (1/3,2/3) can be given position 0.36363636..., etc.

You will need to make some special rules for numbers with more than one expansion (e.g. 0.50000... = 0.49999...) in order to decide where to put them.

[anticorncob6]

I know a situation where the uncountable infinity isn't enough.

Suppose someone shows up and Dave asks, "how many people are at your party?" And he says, "on each person's shirt all the numbers between zero and one are listed, and each number on each shirt corresponds to a digit: either 0 or 1. All possible combinations of zero and one are inside the bus". A similar argument shows that Dave is doomed.

[JoelFeinstein]

@anticorncob6

Yes, the functions from [0,1] (the real numbers between 0 and 1) to the two-point set {0,1} are just the characteristic functions of the subsets of [0,1]. The power set of a set always has bigger cardinality than the original set.

In the March 2010 edition of this talk I called the bigger hotel the Hotel Continuum instead to clarify *which* uncountable infinity was involved.

[anticorncob6]

@TotalGameMaster There is more then one uncountable infinity.

aleph_0 = the number of rooms in the hotel

aleph_1 = 2^(aleph_0)

aleph_n = 2^(aleph_(n-1))

There is no such thing as aleph_infinity, so it would be

limit  aleph_n

n->infinity

which isn't really defined

[TotalGameMaster]

what about the largest number in the universe what would that be

[balachandra108]

@ThePlatinumPengwin

Infinity or beyond infinity can actually be very relevant in our lives. What we understand as our universe is much more than we think or imagine. It shows that we ourselves, may have infinite potential, and possibly be able to come a point of beyond infinity.

[DinoPedia]

Thank you Joel, very informative :)

[MrBradGraham]

basically overall, this just shows how infinity in discrete systems and infinity in continuous systems differ. that is if a rational number can be called discrete! or maybe i am interpreting this wrong? either way its pretty interesting!