Sure, “-∞ < ∞” is a useful concept, but it is not the same thing as when we talk about the sizes of infinities. What we mean by that is how many numbers it contains: (1,2,3,4…) contains fewer numbers than (1.0,…,1.1,…,1.5,…,2.0,…,2.5,…), but how large the actual numbers are, doesn’t matter. The second example contains just as many numbers, is just as “large”, as (1.0,…,2.0).
edit: Sorry for the snarky tone, I was going for nerd maths boy. Hope I at least am technically correct.
But there are infinities which are larger and smaller than other infinities.
-infinity is smaller than +infinity for the most simple example.
Sure, “-∞ < ∞” is a useful concept, but it is not the same thing as when we talk about the sizes of infinities. What we mean by that is how many numbers it contains: (1,2,3,4…) contains fewer numbers than (1.0,…,1.1,…,1.5,…,2.0,…,2.5,…), but how large the actual numbers are, doesn’t matter. The second example contains just as many numbers, is just as “large”, as (1.0,…,2.0).
edit: Sorry for the snarky tone, I was going for nerd maths boy. Hope I at least am technically correct.