(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?

I guess the problem is the starting point, do you start with infinite empty cages that you fill as pets arrive, in which case it can never be full and the question is void, or do you start with an infinite number of cages filled with pets, in which case you won’t ever be able to accommodate extra without changing the rules.

They start filled, but you are wrong to think you can't accommodate another pet while staying in the rule.

I once fell out with a friend about whether there could be different size infinities. I was if the opinion that infinity has a specific meaning and part of that meaning is that it is immeasurable and therefore there can’t be different infinities. He was reading a book that required the philosophical leap of different size infinities existing and was getting very wound up by my refusal to accept it, but couldn’t explain why I should.

I was mostly winding him up. I understood that there are branches of maths that rely on starting from ‘if this imaginary thing existed, then we could do that...’, but it was fun to watch him combust over the fact that I wouldn’t just take his word for it that they did actually exist.

Infinity can always be a bit bigger, or a lot bigger or infinitely bigger and still be infinity. In the cat refuge case you just add a cage at the end of the row, making infinity plus 1 (still equals infinity).

I'm already familiar with this one but infinity does make my head hurt.

Yes there can be...first distinction is countable and uncountable...so for example integers are a countable infinity...but decimals are uncountable...

There are perfectly respectable logical arguments that show if you start with precise definitions of infinite sets and how to compare them, then some are bigger than others. It may be counter-intuitive, but it's a well-established result (and not just in some abstract, imaginary maths sense). But that is where I'm intending to go on this thread with a few more challenges!

Anyone got a solution to the first challenge?...

@BlackAmericanoNoSugar - there is no end cage, that's what infinity means, infinity is not a number and you can't add 1 to it. That's all finite thinking!

Oh and the answer to the cats is move every cat up one cage and put the new cat in cage 1.

If an infinite number of new cats arrive then put existing cats in (cage number * 2) then put new cats in odd number cages!

The Curious Cases of Rutherford and Fry did a podcast on this question, and they were able to logically demonstrate that infinities of different sizes can exist.

Petra - hold that thought!

It’s dh’s Thought but will do...

Yes, Petra has solved the first challenge, and anticipated my second challenge but here it is for those at the back :

Second challenge

The next day, with the refuge still completely full, the inter-planetary dog-catcher turns up and he has been rather too efficient: he has an infinite number of dogs - let's call them D1, D2, D3,... . Can they be accommodated? You can't say move the occupant of C1 up an "infinite number" of spaces - I need to specifically know where to move C1's occupant, so I can program the robot carers.

This is the Hilbert’s Hotel paradox. Hard to get your head around, but fascinating! en.m.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

You need to hope that your cages are arranged sensibly so you don’t have to move too far to your double number...

I love that a question predicated on housing an infinite number of cats is not to be considered abstract or imaginary!

Indeed bookworm - you saw through my thinly veiled disguise.

As Petra has said, the answer to second challenge, is to relocate the occupants like this:
C1 -> C2
C2 -> C4
C3 -> C6
C4 -> C8
...
So all the odd numbered cages will be empty and you can put in the dogs:
D1 -> C1
D2 -> C3
D3 -> C5
...

This shows that the set of even numbers (2, 4, 6, ... ) is the same size as the set of whole numbers (1, 2, 3, 4, ...).

I've got a further challenge - bear with me...

I would say a problem with the original question is once you have taken all of the infinite cats out of the infinite cages simultaneously then you are left with an infinite number of empty cages which you can do anything you like with.

And if you don’t take them out simultaneously then you will be left swapping cats for eternity.

Gotta - it's possible to describe a way to do a process with infinite steps in finite time. en.wikipedia.org/wiki/Supertask

I guess the simplest example is numbers* themselves.

You have an infinite number of numbers. However that must comprise an infinite number of odd numbers as well as an infinite number of even numbers.

*integers ...

As someone who lurks on the active Brexit thread, I'm delighted to see DGRossetti as a guest here. I hope my problems are easier to solve than how to leave the EU!

I’ve reached my limit with that link 😄

This thread has left me with this in my head:

This shows that the set of even numbers (2, 4, 6, ... ) is the same size as the set of whole numbers (1, 2, 3, 4, ...).

This is where I always get confused, because it's so counter-intuitive, even though it makes sense with the dogs and cats example.

And if all the cats then left there would still be infinite dogs so infinity divided by 2 is also infinity.

Spinster - it is counter-intuitive: unlike finite sets, an infinite set can contain a subset that is the same size as the whole set. (Indeed, I believe this is one way of defining an infinite set).

Rutherford and Fry podcasts on the topic of infinity. The OP's talking about Hilbert's Hotel.

BTW, Adam Rutherford is a geneticist not a mathematician (that's Hannah Fry). I'm listening to his audiobook A Brief History of Everyone Who Ever Lived at the moment, and it's excellent.