(Maths) Can one infinity be bigger than another?

[DadDadDad]

Inspired by that circle thread, here's something to get your brains round. The maths in this thread isn't original (and I know there some good mathematicians on MN who will know all this stuff), but hopefully I can give it a new lick of paint with a series of challenges.

You are the manager of the best pet refuge in the universe - its cages are labelled C1, C2, C3, ... with no limit - it's infinite! The rule is that each cage cannot have more than one occupant. Then again, why would you when you have unlimited supply?

Fortunately, you have an infinite set of robot carers who can feed, walk, clean etc the occupants of every cage.

First challenge:

One evening, the refuge is full: each of C1, C2, C3, ... is occupied. You are about to lock up when someone turns up with a little kitten needing a home for the night. You would love to accommodate it - but how?

Like I say, this is a well-worn path for mathematicians. For those still interested, it's gets a bit harder...

Third challenge.

The refuge is completely empty, all cages unoccupied. Then, to your horror, a spacecraft lands carrying a infinite collection of Neptunian fractional elephants. These are elephants that can be any size, so an infinity of them can be carried in a finite spacecraft. The astrozoologist who rounded them up has so many, she has labelled them with fractions - see picture. Every positive number you can express using fractions (a whole number divided by a whole number) that you can think of has an elephant (including the whole numbers 1, 2, 3, ...). Can you accommodate them? How can I program the robot carers to systematically house them all? Or is this set too big?

Yes you can, it’s a countable infinity. You imagine a grid that defines all your fractions (integers along top and down side and then grid entry is top number/side number) then you can work out a pattern that starts at top left (1,1), then draws a line going through all the diagonal lines (so from 1,1 to 2,1 to 1,2 to 1,3, 2,2, 3,1 etc)...then you can predict the position of any fraction in the grid based on how far along this set of diagonal lines you are (I’d need pen and paper to work out the formula)...then that gives you the cage number...

Guess who is married to a mathematician!!

Textbook answer. My DW is married to a mathematician but wouldn't know of any this...

Tbf I like maths too - he’s just the one who has the degree in it!

To set out Petra's solution in a slightly more concrete way:

Express every number as a fraction in the form a/b. So 3 is 3/1, 13 2/3 is 41/3, etc. Now, organise the elephants like this...

First, line up those elephants whose numerator and denominator add up to 2. Actually, there's only one: 1/1. It goes in C1.

Now, line up those elephants whose numerators and denominators add up to 3: 1/2 and 2/1. They go in C2 and C3.

Then, those adding to 4: 1/3, 2/4, 3/1 - into C4, C5, C6. You might notice that 2/4 is the same as 1/2 and it has already been allocated. That hardly matters, just leave a cage empty. The point is we will systematically work through every fraction and find them a cage.

What this shows is that the set of all fractions (the rational numbers) is the same size as the set of whole numbers - against intuition, the two can be paired up.

For Petra, if you want a formula for what I've described: we can say that cage number 1+(n-1)(n-2)/2 will contain elephant 1/(n-1), and the cages consecutive to that will contain 2/(n-2), 3/(n-3) all the way up to (n-1)/1. (But leaving a cage empty if the fraction cancels down to something that's already been allocated).

One more challenge to come...

You must have an infinite number of empty cages within that presumably!!!

Ordinal infinities are fun too! In that setting, 1 + infinity is the same as the infinity you started with, but infinity + 1 is bigger.

OK - I think this is the last one.

Fourth challenge

After you manage to pass on to new homes all those fractional elephants, you have barely let out a sigh of relief that the refuge is quiet again, when a hyper-dimensional star-cruiser arrives carrying an infinite collection of Denebian ring-tailed mega-lemurs.

These lemurs are a mathematical marvel. They have tails with an infinite number of rings (each ring is half the width of the one before it so they can fit on a finite tail). Owing to some quirk of evolution, numbers appear on their fur with each ring showing a digit from 0 to 9. The alien flying the ship is able to confirm that every possible sequence is represented in its collection: there's a lemur which is 5-0-0-0-0-0-0...; there is a lemur where the prime number rings are a 7 and the others are 3; there is a lemur which reads 3-1-4-1-5-9-2-6-... who for obvious reasons is nicknamed pi; there's one whose digits read out the mobile phone number of every MNetter, then repeat. It is quite the multitude.

How can you organise them to fit them in the cages?

Infinity isn’t a number.

It can’t be the case that all the cages are full - there’s no such thing as “all”.

An infinite number of cages means an infinite number of pets can be accommodated. So you’re asking a bad and irrelevant question.

All of these are bad questions, OP. They seem to be operating on the assumption that infinity means “very, very big”. It doesn’t.

Everything can be accommodated, no matter how they are labelled.

Infinity is a concept rather than a number.
What's infinity plus 1? Infinity.
Infinity to the power of infinity? Still infinity.

Therefore you can have different sizes.

You can’t put your lemurs in cages...they are an uncountable infinity and your cages are countable....

Proof is put all lemurs in a line
Take the first digit from the first lemur and add 1 gives you first digit of new number
Take the second digit from the second and add 1 gives second digit...
Etc

Your new number must be different to all other lemur numbers because it is always different in one digit from any of them...

Hence uncountable infinity...

Oh and none of these questions relies on infinity being “just a very big number”...they all rely on it being infinity.

Dh says what do you do with the infinite amount of animal poo???

I know infinity is not a number, that is why I am definitely not treating it as very big number. None of these problems would work in the same way for a very big number.

The cages and pets are just a way to visualise it. If you forget about them, then there is no doubt that there are infinite sets (eg the set of whole numbers) and it can be proved that sets which are apparently different in size are in fact the same size.

I don't think these can be "bad and irrelevant questions" when they've been posed and solved by some of the great mathematicians, starting with Galileo, with great breakthroughs coming from Cantor and others.

en.wikipedia.org/wiki/Galileo%27s_paradox
en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article

I have severe dyscalculia and this thread is making me want to cry...

Sorry, Jarets - this thread is just a bit of fun, not intended to make anyone feel bad. I did put "Maths" in the thread title as a fair warning!

The answer to your question is in the detail....
Each cage has a robot in it...
But does the cage already have a dog in it?
If so then the cage will always be full.... But it also means there is no room for anything in the world except the dogs and cages, so where did the dog catcher come from?

The robots are outside the cages as they are programmed to move the pets around.

You can an infinite set which takes up finite space. If the elephants have masses of 1 kg, 1/2 kg, 1/4 kg, 1/8 kg, ... then their total mass is 2 kg, even though the set of elephants is infinite.

This is all very well, D^3, but if my DD2 has an infinite number of ways of moaning about schoolwork, how can I get her to do maths revision?

Unfortunately, maths depends on logic which I don't think reliably operates when it comes to the emotions of teenagers.

I'm enjoying this thread, even though I've been reasonably well schooled on infinity. It's a lot easier to read it than have a teenager educate you verbally at high speed about it!

Thank you, Kitten. Petra has solved my fourth challenge, but you've encouraged me to put my own explanation of why the lemurs can't all be accommodated, and so they represent a bigger set than all the preceding cases.

Suppose your talented, but inexperienced deputy manager thinks they've succeeded in caging the whole lot: the star-cruiser looks empty and every cage contains a lemur. Now consider the following as an example:
Suppose the first ring of the lemur in C1 is 2 - you write down 3.
Suppose the second ring of the lemur in C2 is 8 - you write down 9.
Suppose the third ring of the lemur in C3 is 9 - so now you write down 0.
Repeat for fourth ring in C4, and so on - always writing down a digit that differs.

Now think about the sequence that you have written down. Let's say it's 3-9-0-...

There must exist a lemur with that sequence because you were told the collection was complete - let's call it lemur X. But which cage is it in?

It is not in C1, because X has 3 on it but C1 has 2.

It is not in C2, because X has 9 and C2 has 8 on their second rings. Similarly, X is not in C3, C4 and so on - whichever cage you compare with there is at least one ring that differs. Your deputy manager has missed lemur X out - and is always doomed to miss some. (In fact, he will miss infinitely many).

Dad I’m only joking! I find it fascinating even though I’m struggling to follow it at all

There is an easy way to do this.

Build a new level for each animal ...