Funnily enough that’s an example of two infinities that mathematically have the same cardinality (which is very often conflated with size since for general mathematical purposes that’s what it is) since you can map a bijection (i.e. every number in the first set has one and only one mapping in the second and vice versa) between the two (and it’s as simple as f(n)=2n).
And intuitively that makes about as much (or rather, little) sense as the infinite hotel.
An example of infinities with different cardinalities would be rational numbers vs natural numbers.
I didn’t actually know this (all my math knowledge comes from what my cs degree forced me to ingest) but google says yes (since natural numbers, being a countably infinite set, are apparently an example of the smallest possible cardinality of infinite sets, so any infinite subset of natural numbers is always the same cardinality.)
So, some infinities are bigger than others.
How many numbers are there? An infinite number.
How many even numbers are there? An infinite amount, but half the size of the first infinity.
This is how there are empty rooms in the infinitely large hotel with infinite guests.
Funnily enough that’s an example of two infinities that mathematically have the same cardinality (which is very often conflated with size since for general mathematical purposes that’s what it is) since you can map a bijection (i.e. every number in the first set has one and only one mapping in the second and vice versa) between the two (and it’s as simple as f(n)=2n).
And intuitively that makes about as much (or rather, little) sense as the infinite hotel.
An example of infinities with different cardinalities would be rational numbers vs natural numbers.
OK, thanks for the extra info. When I see that function it kinda makes sense but stops if I think about it too much.
Would even numbers and prime numbers have different cardinality?
I didn’t actually know this (all my math knowledge comes from what my cs degree forced me to ingest) but google says yes (since natural numbers, being a countably infinite set, are apparently an example of the smallest possible cardinality of infinite sets, so any infinite subset of natural numbers is always the same cardinality.)
The answets to this stackexchange question go into detail but that’s beyond what I understand without a lot of effort tbh.
OK, that makes some degree of sense in the abstract. I’ll try to hold on to it.
Thanks for checking and getting back to me. It helps grok the cardinality a bit more.
Half of infinity is still infinity, but smaller?
Can you hand me that spatula? I need to scrape my brains off the walls.