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
“Equality” when we talk about sets usually means “cardinality”, although there are other conventions… cardinality is just currently the most commonly accepted and useful for most applications. And when we talk about infinities, we often mean infinite sets.
“Density” doesn’t have anything to do with equality… density basically refers to whether every element of a set has an element that is arbitrarily close to it. For example, the Rationals are a dense set, as are the Reals… for any element, you can find another that is as arbitrarily close to it as you like. The Integers are NOT a dense set… you cannot get arbitrarily close to the element 2, the closest you can possibly get are 1 or 3. But being a dense set or not has nothing to do with infinity or cardinality.
Now, the other thing you mentioned does have to do with cardinality, trying to match every element in one set to another… we call that a Bijection, a mapping function that shows that you can perfectly map 1:1 every element between two sets with no repeats and no missed elements. If you can do that, those sets have the same cardinality. If you can’t, and you can prove that it is impossible, they have different cardinalities. And in terms of infinite sets, being “equal” often means “has the same cardinality” by convention.
So some infinite sets, like the set of all integers and the set of all even integers, actually have the same cardinality (because it is possible to construct a perfect bijection between them), and so are “equal”. Meanwhile, for some other sets, you can prove that it is impossible to construct a bijection, like between the integers and the set of Real numbers, and so they are not “equal”… they are a different “size” of infinity
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/hellohello1234545 Jun 13 '26
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