gatekeeping

Infinity gangsta when aleph null walks in

What about infinity times zero?

Oh yeah? What's infinity divided by zero?

Quaternions hello?

What about quaternions and octonions and ...

That leaves out quaternions

This image goes so hard

Thanks.

I mean, whoever gets to declare a "last number" that works certainly will get some bragging rights. After all, you can only ever declare one.

...Right?

(I know math is very weird)

Which of the infinities? There are many, many :D

Oh no! Please don't tell me there are infinity infinities!

Wait, they ran out of greek letters and started using Hebrew ones now? When did that happen?

No matter what Wikipedia says, Aleph Null is the real way to say it, because it sounds so much cooler

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 are all bigger infinities. Furthermore, there are multiple kinds of transfinite algebra. Cardinal algebra behaves mostly like how you described, except every transfinite cardinal has a 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.

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 there are numbers 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 embedding 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.

Is infinity to the power of infinity special?

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.

Edit: In modern systems aleph_0 = omega btw. Omega denotes ordinal and aleph denotes cardinals.

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?

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!

From gatekeeping to gaslighting in 2 comments. Not bad!

Here's a thousand page proof defining all the logical underpinning required to prove that 1 + 1 = 2.

Sad Frege noises

That regex implies “” is a number

I didn't realize '.' is a number.

\([0-9]+\.[0-9]\)?[0-9]* is more accurate I think.

well sure, if you want to be fancy. i was speaking in layman terms for the rest of the world.

regex for the win.

yeah but by Cantor's diagonal argument, you still wouldn't be listing all the real numbers

What about imaginary numbers?

UΩU

g

But those are parentheses, are they not? I was taught intervals using square brackets and semicolon. While parentheses are used for coordinates and tuples. The square brackets indicates inclusion of the boundary number.

Ie. the statement "2

Update: apparently either lemmy or my app (boost) wasn't that excited for my less than signs, and just skipped the rest of the comment. And here I had spent time copying both "less than or equal to" and infinity signs, since my keyboard doesn't seem to have them... For the time being pls disregard the comment above, while I figure out how to write math on lemmy.

That was the root of what I was getting at, thanks lol.

e