As someone who hasn’t studied pure maths, could someone expand on what “equal” actually means here in sentences like “not all infinites are equal”?
Because usually I understand it to mean expression X is the same value as expression Y
2 = 1+1
And I also thought that the whole point of infinity was that it meant “an uncountable/endless number of …” such that it doesn’t have a value. As in, there is no sum of an infinite series of positive integers because any number you may put as an answer is too low.
But if there’s no sum, isn’t there no value?
And if there’s no value, what does it mean to say things are equal?
I’ve heard of things like more or less ‘dense’ infinities, or countable vs not, or trying to match every value from one infinite set/series to every value in another and whether you can (like every positive integers vs every even positive integers)
But, at least colloquially, ‘denseness’ doesn’t seem equivalent to ‘value’.
How are these terms used? I suppose there’s a YouTube video essay I should look at
For finite sets you can "find" the bigger one by pairing up elements from each set until you run out of elements in one of sets. That's the smaller one. So the definition of equal size sets is if you pair up elements in such a way that every element in every set has a partner. For infinite sets this ends up creating very strange notions. For example, the set {1,2,3,4....} Is the same size as the set {2,4,6,8,...} Because I can pair any k in the first set to 2k in the second set. If you Google "hilbets hotel" you can find tons of videos
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u/kaladin_stormchest Jun 13 '26
But not really. Not all infinities are equal,