takes a big swig of Cantor depends if you’re talking ordinals or cardinals, my dude. Let’s get heavy with infinities my man.
TL;DR you can’t use infinity like that and your calculus professor will yell at you if you try.
Infinity isn’t a real number and it’s not generally useful to think of it as one like the dude in this comic is trying to. However, in calculus you can treat it as a concept that a variable or expression can approach. In that way, “approaching infinity” is just another way of saying “increasing forever” or “given a number x, you can always use x+1”. This is why expressions like “infinity = infinity" or “infinity = infinity+1” like the comic are not useful statements.
That’s also why your calculus professor is so insistent that you write out the whole limit notation, because it’s nonsense to just throw infinity into an expression raw (like “infinity+1” in the comic). But, if you think of it as “the limit of x+1, where x approaches infinity”, then it’s clear that infinity doesn’t have anything to do with the actual values, it’s just used to describe potential values.
Here’s an example if that still doesn’t make sense: Bob and Jill are twins who were born with 0 and 1 dollars respectfully, but both earn a dollar a day forever because they’re immortal. Just because they will live forever, doesn’t mean that they’ll ever be able to say “I’m infinite years old”. They’ll always be x years old, but x will increase by one every year from their birthday for the rest of time. For the same reasons, they’ll never be able to say “I have infinite money”, but if they don’t spend it, it will increase forever, approaching infinity. And finally, if neither Bob or Jill spends anything and that dollar a day is their only income, then Jill will always be worth a dollar more than Bob, even though both have infinite wealth potential.
the concept of infinity + 1 can be rigorously defined (as an ordinal number). the basic idea is that infinity +1 is the set containing every single positive whole number, in increasing order, and then something else.
but what you said about infinity in calculus is correct. the “infinity” that appears in calculus is conceptually a different idea of infinity and it’s basically just an inconvenient choice of notation that they’re called the same thing.
in complex analysis, there’s also the riemann sphere, which is basically a way to view the sphere as the complex plane in addition to the “point at infinity”. i.e., 0 is the south pole, and infinity is the north pole. and in this context it’s fairly common to say stuff like “f(infinity) = 0” or “f(2) = infinity”. these can all be understood in terms of limits as you described, but it does sort of blur the line between “actual value” and “potential value”, since infinity is actually a point in the riemann sphere, but it’s primarily described in terms of limits.
That’s why my coffee shop hands out Gabriel’s Horn shaped cups. Bottomless, but finite volume. What a scam.
I prefer my coffee in a Klein bottle. I only drink ∞ a day
Do you want small, moebius or large?
I zeno problem with his paradox.
The symbol of infinite is a stand in for a constraint, or an idea. That’s not the same thing as the real world expression or experience of that idea.
And that’s why nothing in nature is infinite. Except human stupidity if you want to believe Einstein.
yet the universe is probably infinitely large, and expanding.
Oh, I believe him. Self-evident.
Relative to me you’re pretty smart though.
But there are infinities which are larger and smaller than other infinities.
-infinity is smaller than +infinity for the most simple example.
To add to what Kabi said, IIRC only when you’re speaking in set or groups do the infinities become “larger” (simplified and not 100% accurate). I.E. infinity of regular numbers vs infinity containing all the variations of positive integers added. The latter would be “larger” cause it contains multiple infinities or “sets” of infinities and is infinite within itself. This video helps explain probably better
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.
Yeah, I was going for simple rather than correct. I didn’t want to get into explaining Cantor’s Diagonalization to Lemmy folk.
