Beyond Infinity? (Part II)

[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.

[TotalGameMaster]

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

[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

[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!

[semasiologistics]

I did not like this video because it was much too complicated for "laymen".

[JoelFeinstein]

Did you try Part I?

The first part is more suitable for laymen. The second part is definitely tougher. Joel

[semasiologistics]

I saw both, thank you. Yeah, the first is really easy, you are right. It is just the second that seems to forget that the rest of the world is not conversant with the "Nth" place etc. Terrible job in that respect. Also, did not make well on the "beyond" bit. ..

[JoelFeinstein]

Yes, the second half is intended for people with a bit more technical knowledge. It is probably best for first-year maths undergraduates, or possibly advanced maths A level students. The professional mathematicians at the time also had trouble accepting it, so it is not surprising that it causes trouble for the layman!

[ThePlatinumPengwin]

as a mathmatical and scientific theorist, this to me is relatively meaningless, but only because there cant be practical application involving infinity, because being infinite it will always be unattainable.

[yewklid]

What is a mathematical and scientific theorist?

What does "being infinite it will always be unattainable" mean?

[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.