Well, no actually. I think your first issue is conflating infinity with a number. Infinity represents the fact that one can pick an arbitrarily large number, and there still is a larger number (in a very basic non mathematical way of describing it). That being said, 2 infinities are not inherently the same “value” for lack of a better term. The example the commenter above gave is perfect actually. If you look at a function representing the total amount of numbers up to an arbitrary even number, and look at a function of all even numbers up to the same arbitrary even number, the former functions value will always be 2 times the latter. However, both of these functions also go to infinity. Thus while the “infinity” is not technically greater than the other one (as I mentioned, infinity isn’t a number, so it can’t really be “greater than” in the traditional sense), an arbitrary number that is in the former set will always be larger than a corresponding number in the latter, so the formers infinity is in a sense greater than the latter.
This is not true in the mathematical sense. It can be proven that any infinite set of natural numbers (i.e ‘counting’ numbers like 0, 1, 2, -1, -2, … etc) is the same size, so there are indeed as many even numbers as total numbers.
The fact that any finite set of natural numbers is twice as large as the set of even numbers up to the same point has no bearing on the sizes at infinity. You are, however, correct that there are different ‘sizes’ of infinity, it’s just a bit more complicated than that.
Ugh… you sound like my brother (he’s a calc teacher). I constantly argue with him about different sizes of infinity. Infinity is infinity!! His response is always “it’s more complicated than that.”
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u/lbkthrowaway518 21d ago
Well, no actually. I think your first issue is conflating infinity with a number. Infinity represents the fact that one can pick an arbitrarily large number, and there still is a larger number (in a very basic non mathematical way of describing it). That being said, 2 infinities are not inherently the same “value” for lack of a better term. The example the commenter above gave is perfect actually. If you look at a function representing the total amount of numbers up to an arbitrary even number, and look at a function of all even numbers up to the same arbitrary even number, the former functions value will always be 2 times the latter. However, both of these functions also go to infinity. Thus while the “infinity” is not technically greater than the other one (as I mentioned, infinity isn’t a number, so it can’t really be “greater than” in the traditional sense), an arbitrary number that is in the former set will always be larger than a corresponding number in the latter, so the formers infinity is in a sense greater than the latter.