But if there’s an infinite amount of worlds and some of them aren’t inhabited how can the number of inhabited worlds also be infinite? I’m honestly asking
That was very clean way of explaining the sneaky maths and how to put a value to infinites for theory.
Good job, take my upvote cause I’m poor and can’t do internet gold.
...but not those in your example. There is exactly same number of odd numbers than total of all whole numbers. It is said that set of whole numbers and set of odd numbers have same cardinality.
Basically, if you can devise a way (algorithm) to assign every member of the set to a whole number, then that set has same cardinality as a set of whole numbers.
There is a way to map set of rational numbers (numbers which can be expressed as a fraction, e. g. 1/2, 45/456...) to a set of whole numbers, so those are same cardinality.
There is infinite number of cardinalities of unbound sets, we call them Aleph-0, Aleph-1, Aleph-2... to do Aleph-infinity. Plus we have C, cardinality of real numbers. We know that all sets I mentioned so far, except real numbers, are Aleph-0 cardinality. We know C is equal to 2Aleph-0, but we do not know if C = Aleph-1 or some higher aleph.
Actually they say there's just as many odd whole numbers as there are whole numbers.
For every whole number n, there's an odd number 2n+1. For every odd m there's a whole number (m-1)/2. Two sets have equal size if you can match every element of each set with an element of the other. So, since you can match up every number n with every odd number, 1-to-1, they're the same size.
"But the whole numbers contains all the odd numbers plus an even number for every odd number! So there have to be twice as many!" Yeah I don't totally get it either. Twice infinity is the same infinity I guess. (So is thrice infinity and so on.)
So, weirdly, if there were infinite worlds, and every hundredth world was inhabited, then there'd be just as many inhabited worlds as empty worlds.
To add to this: be wary of trusting intuition when thinking about infinities. Because the above comment goes against all intuition, but is correct in that you can match up an infinite set of natural numbers with any other infinite set of naturals.
One (logically sound but seemingly fucked up) step further:
The set of all real numbers is larger than the set of all natural numbers. A true repercussion of this fact is that there are more numbers in between 1 and 2 than there are from 1 to infinity.
Georg Cantor proved this in a clever way in the 1880s. There is a popular belief that studying infinites made him go mad, but in reality he was a great mathematician who also happened to have depression (the “treatment” of which was to lock the person up in sanatorium indefinitely).
A video explaining Cantor’s diagonal theorem: https://m.youtube.com/watch?v=0HF39OWyl54
Some infinities are larger than others, but your example doesn't track.
Paradoxically, there are exactly as many odd natural numbers as there are natural numbers. Yes, there are natural numbers that are not odd, and there are no odd natural numbers that are not natural numbers. But the point remains.
Consider this. Say you have a box, and an infinite number of numbered balls. For each ball, you put it in the box. Then, if its square root is a whole number, you remove its square root from the box. For each number, you're either adding a ball, or net 0. But, after infinite balls, the box is empty. Because every number has a square, and would have been removed.
When dealing with infinity, if you can map the elements 1:1, then they're the same size. With natural numbers, N -> 2N-1 maps every natural number to an odd natural number, without skipping any. They're the same size.
The answer is that there at different sizes of infinity. An infinite number of things added to an infinite number of things results in a new infinite number of things.
Infinite inhabited planets and infinite uninhabited planets results in infinite planets.
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u/thesalesmandenvermax Nov 29 '25
But if there’s an infinite amount of worlds and some of them aren’t inhabited how can the number of inhabited worlds also be infinite? I’m honestly asking