Infinity - 1

[Cabbage]

Harmonix:

Ok, I can see how removing an infinite subset from an infinite set results in an infinite set.

Well, that’s not really true. Removing an infinite subset from an infinite set can result in an infinite set, but not necessarily. For example, take the reals (an infinite set), remove all but the number zero (so we’re removing an infinite subset), but the result is the set {0}–a set consisting of just one element.

I do not however see how you can remove something from set 1 and have the resulting set 2 be exactly the same as set 1.

You’re absolutely right here–set 1 and 2 will be different sets, however, you seem to be missing the point we’re trying to make, and that is that alhtough the two sets are different, the sets will be the same size (cardinality).

For a very simple example, the sets {1,2,3} and {4,5,6} are completely different sets, but both are the same size (each consisting of three elements).

Similarly, take the reals, and remove the number zero. The resulting set is not the same as the reals (it’s missing zero!), but it is the same size (cardinality) as the reals! This is a feature unique to infinite sets–you can remove a finite number of elements, and the resulting set will have the same cardinality you started with! You can’t do this with finite sets. Intuitively, you can think of this as saying that infinite sets are so big, throwing out finitely many elements has no effect on the size of the set.

I can however see how the set of natural numbers is exactly equal to the set of irrational numbers.

I believe you meant rational numbers. The set of irrational numbers is much bigger than the set of natural numbers.

don’t infinities exist?

limit x - > 0 1/2x
is different from
limit x -> 0 1/x

Yeah, infinities exist (as in the cardinalities I was referring to above), but not in the sense you mention here. Neither of these limits exist–we say the limit is plus/minus infinity (depending on whether you approach zero from the right or left), but that’s really just a specific way of saying how the limits fail to exist. It’s just a shorthand way of saying that the function grows without bound as x approaches zero, and has nothing to do with the cardinalities of sets that I’ve been talking about.

[green_dragon_1]

I always liked the “hotel with infinite rooms” as an illustration of how infinity behaves when you start doing arithmetic on it.

Suppose there is a hotel with infinte rooms, and no vacancies. Moreover, the hotel owner is very greedy and so will do everything he can to make more money (yes I know that he’s already making an infinite amount of money, but he wants more. Imagine, if you will, that Bill Gates owns this particular hotel :)).

Now let’s look at a couple of scenarios:

1. A single person turns up wanting a room; what does the hotel owner do?
   He moves the person in room 1 to room 2, the person in room 2 to room 3 etc, throughout the hotel. For any room number you can think of, they just get moved up one and as there are infinite rooms, there will always be a room for them to move into.
   The end result is that room 1 is now empty and so the new lodger has somewhere to stay. Effectively, we have just added one to infinity; the answer is of course just infinity because the number of rooms in the hotel has not changed.

2. To illustrate infinity-1, just imagine that the guest in room 1 leaves and every other guest moves down one room number; the logic is essentially the same, every room will be filled and the number of rooms has not changed, so infinity-1=infinity.

3. Suppose an infinity of guests turns up (the hotel manager jumps for joy!). To be able to accomadate them all, the hotel manager moves all the guests to the room which is twice the room number of their original room, so:
   guest in room 1 -> room 2
   guest in room 2 -> room 4
   guest in room 13 -> room 26 etc.
   Now all the odd numbered rooms are free. There is an infinity of odd numbers (in a hotel with infinite rooms you can always find a bigger odd number) and so the infinite guests can now move into all the odd numbered rooms.
   So infinity+infinity=infinity

This example did the trick for me, hope it helps others to think about it.

Note: there are operationswhich will give you different kicks of infinity, e.g. get you from the countable infinity (cardinality of the integers) to the uncountable infinity (continuum of the reals), but not using any of the basic arithmetic operations we’re talking about here.

Cheers,
DC

[green_dragon_1]

:smack: different ‘kicks’ of infinity? I meant different ‘types’ of infinity:)

[Mangetout]

*Originally posted by green_dragon *
3. Suppose an infinity of guests turns up (the hotel manager jumps for joy!). To be able to accomadate them all, the hotel manager moves all the guests to the room which is twice the room number of their original room, so:

The downside is that organising this occupies an infinite amount of the receptionist’s time and she gets paid time and a half for anything over 37 hours a week.

[green_dragon_1]

Alright, let’s just have an omnipresent, omnipotent hotel receptionist so it in fact takes him/her barely any time at all, and the hotel manager can in fact just send her home early

Glad that’s the only flaw you found with my logic though!