IIRC Depends if you talk about cardinal or ordinal numbers. What I remember: In cardinal numbers (the normal numbers we think of, which denote quantity, etc.) have their maximum in infinity. But in ordinal numbers (which denote order - first, second, etc.) Can go past infinity - the first after infinity is omega. Then omega +1. And then some bigger stuff, which I don’t remember much, like aleph 0 and more.
Comment on gatekeeping
veniasilente@lemm.ee 6 months ago
Aren’t there numbers past (plus/minus) infinity? Last I hear there’s some omega stuff (for denoting numbers “past infinity”) and it’s not even the usual alpha-beta-omega flavour.
Come to think of it, is there even a notation for “the last possible number” in math? aka something that you just can’t tack “+1” at the end of to make a new number?
KmlSlmk64@lemmy.world 6 months ago
weker01@feddit.de 6 months ago
No cardinal and ordinal numbers continue past the “first” infinity in modern math. I.e. The cardinal number denoting the cardinality of the natural numbers (aleph_0) is smaller than the one of the reals.
veniasilente@lemm.ee 6 months ago
So wait, you can’t have numbers larger than infinity, but you can order them “past infinity”? I’m trying to wrap my head around the concept, and the clearest thing I can get at the moment is that the "infinity+1"th number is infinity… would that be right?
weker01@feddit.de 6 months ago
No you can have numbers past infinity op is wrong.
As for how to order past the first infinity it’s easy.
Of course first you have 1 < 2 < 3 < 4 < … Then you take a new number not equal to any of the others let’s call it omega. Define omega to be larger than the others. So 1 < omega, 2 < omega,…
This you can of course continue even further by introducing omega + 1 which is larger than omega and therefore larger than all natural numbers.
You can continue this even further by introducing a new number let’s call it lambda that is bigger than all omega + x where x is a natural number.
This can be continued forever i.e. an infinite amount of times.
Leate_Wonceslace@lemmy.dbzer0.com 6 months ago
Hi! I’m a mathematician, and if you want to know more about infinity, I recommend this video: youtu.be/23I5GS4JiDg
veniasilente@lemm.ee 6 months ago
After reading how this thread is going I’m half expecting this to be a Kurzgesagt video or something equally “cutesy existential dread” inducing lol. Let’s see what do I find!
jflorez@sh.itjust.works 6 months ago
There is nothing past infinity on the real number line. Then there is the imaginary line that gives you an infinity for the complex numbers
kerrigan778@lemmy.world 6 months ago
There is nothing “past” infinity, infinity is more a concept than a number, there are however many different kinds of infinity. And for the record, infinity + 1 = infinity, those are completely equal. Infinity + infinity = infinity x 2 = still the same kind of infinity. Infinity times infinity is debatably a different kind of infinity but there are fairly simple ways of showing it can be counted the same.
Leate_Wonceslace@lemmy.dbzer0.com 6 months ago
Hi, I’m a mathematician. My specialty is Algebra, and my research includes work with transfinites. While it’s commonly said that infinity “isn’t a number” I tend to disagree with this, since it often limits how people think about it. Furthermore, I always find it odd when people offer up alternatives to what infinity is; are numbers never concepts?
Regardless, here’s the thing you’re actually concretely wrong about: there are provably things bigger than infinity, and they all are bigger infinities. Furthermore, there are multiple kinds of transfinite algebra. Cardinal algebra behaves mostly like how you described, except every transfinite cardinal has successor (e.g. There are countably many natural numbers and uncountably many complex numbers). Ordinal algebra, on the other hand, works very differently: if ω is the ordinal that corresponds to countable infinity, then ω+1>ω.
CompassRed@discuss.tchncs.de 6 months ago
You have the spirit of things right, but the details are far more interesting than you might expect.
For example, there are numbers past infinity. The best way (imo) to interpret the symbol ∞ is as the gap in the surreal numbers that separates all infinite surreal numbers from all finite surreal numbers. If we use this definition of ∞, then every infinite ordinal number is greater than ∞. For example, every infinite surreal number is greater than ∞ by the definition of ∞. Furthermore, ω > ∞, where ω is the first infinite ordinal number. This ordering is derived from the imbedding of the ordinal numbers within the surreal numbers.
Additionally, as a classical ordinal number, ω doesn’t behave the way you’d expect it to. For example, we have that 1+ω=ω, but ω+1>ω. This of course implies that 1+ω≠ω+1, which isn’t how finite numbers behave, but it isn’t a contradiction - it’s an observation that addition of classical ordinals isn’t always commutative. It can be made commutative by redefining the sum of two ordinals, a and b, to be the max of a+b and b+a. This definition is required to produce the embedding of the ordinals in the surreal numbers mentioned above (there is a similar adjustment to the definition of ordinal multiplication that is also required).
Note that infinite cardinal numbers do behave the way you expect. The smallest infinite cardinal number, ℵ₀, has the property that ℵ₀+1=ℵ₀=1+ℵ₀. For completeness sake, returning to the realm of surreal numbers, addition behaves differently than both the cardinal numbers and the ordinal numbers. As a surreal number, we have ω+1=1+ω>ω, which is the familiar way that finite numbers behave.
What’s interesting about the convention of using ∞ to represent the gap between finite and infinite surreal numbers is that it renders expressions like ∞+1, 2∞, and ∞² completely meaningless as ∞ isn’t itself a surreal number - it’s a gap. I think this is a good convention since we have seen that the meaning of an addition involving infinite numbers depends on what type of infinity is under consideration. It also lends truth to the statement, “∞ is not a number - it is a concept,” while simultaneously allowing us to make true expressions involving ∞ such as ω>∞. Lastly, it also meshes well with the standard notation of taking limits at infinity.