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99999...9^9^9^9^9^9^9...9!!9!!9!!9!!9...9↑↑↑↑9↑↑↑↑9↑↑↑↑9∞Depends on what is allowed ig
Small note,
9↑↑↑…↑↑9
Would be
near infinitelymuch much much larger than just repeated hexation of 9(Of course even just 9↑↑↑↑9 is too big to be written in the universe, so all of these numbers are practically a microdose of infinity)
It’s not “near infinitely larger” since there are a finite number of numbers before it and an infinite number after it - it’s nowhere close to “near infinitely larger”
∞ ↑↑↑↑↑… ↑↑↑∞
Fair enough
I would say it’s incomprehensibly larger, though, which is what I took your meaning for in the first place.
Infinty is a magnitude, not a number
Also BB(8000) can (proven ) not be represented by ZFC so that might take the cake
If we get to bring infinity into the mix, how about larger infinite number sets? Aleph-omega time.
infinity is great indeed
∞
Infinity isn’t a number, it’s a concept. Some infinities are bigger than others. For example, there’s an infinite amount of real values between 0 and 1 but there’s an even greater infinite amount of real values between 0 and 2.
May as well go through the proofs:
First, we need to establish that two infinities are equal in cardinality (aka size) if all their elements can be 1:1 mapped to each other.
So, to go from the reals within [0, 1] and [0, 2], we can multiply by 2. This maps every value within [0, 1] to every value within [0, 2], so these are of the same cardinality.
Where things get interesting is the proof that the reals within [0, 1] are of greater cardinality than every integer.
Say we have an arbitrary mapping from every integer to a real within [0, 1]:
0 -> 0.89236… 1 -> 0.47389… 2 -> 0.84776… 3 -> 0.18790… 4 -> 0.90542… ⋮ ⋱This list contains every integer, but it does not contain every real number because we can always come up with a new one by ensuring at least one digit is different in each existing real:
0 -> …8… ≠ 9 1 -> …7… ≠ 8 2 -> …7… ≠ 8 3 -> …9… ≠ 0 4 -> …2… ≠ 3 ⋮ ⋱ 0.98803… is not within the listTherefore, no 1:1 mapping between the integers and reals exists. Because the limiting factor is the amount of integers, the cardinality of the reals is greater than that of the integers.
Edit: https://en.wikipedia.org/wiki/Cantor’s_diagonal_argument
That Vsauce video really has done some damage, huh
The smallest infinity is the countable infinity. It is the cardinality (think ‘size’) of the natural numbers (1,2,3,4,…), hence the name.
Unintuitively, the whole numbers (Natural numbers, 0, and Negatives) have the same cardinality. That means you can match up each natural number with a whole number one-to-one. (‘there exists a bijective function’)
Even stranger, the rationals (-½,1.3,16.6…) also have the same cardinality as the naturals. The proof is a bit more involved, but still not that hard.
Now, what infinity is larger than others, then? This is where we find the Reals (non-terminating decimals, π, e, √2). No matter what you do, you cannot match them up with the naturals. If you’re curious about that, look up Cantor’s diagonal argument.
But, interestingly enough, the numbers between 0 and 1 have the same cardinality as the Reals! Any interval within the Reals is the same ‘size’ of infinity as the entire Reals. You can always find a one-to-one correspondence between the two. (For (0,1) and R you could pick tan, for example)
More generally, if you want to produce a ‘larger’ cardinality from an existing infinite set, you can look at it’s power set. That’s the set that contains all possible subsets from the original, and always has a larger cardinality than the old one.
“Infinity” and “number” mean different things in different contexts. In the context of set theory, its perfectly valid to talk about infinite numbers, e.g. https://en.wikipedia.org/wiki/Aleph_number
Those aren’t infinitely large numbers as they aren’t the same ordinals as regular real numbers. When talking about large numbers it’s implicitly assumed that they can be compared to real numbers.
In other words: Is 1 second bigger than 1 liter?
Those are very explicitly not referring to “the number infinity” though. They’re cardinalities. A number is used to represent the cardinality of the set, but that number is not the same as the set. It refers to the size of the set, not the value of the set. Many other sets could have the same cardinality.
I’m not quite sure what you mean by “value of the set”. Do you mean ordinal numbers? Yes, those are distinct from cardinal numbers, but they’re both referred to as “numbers” in set theory. My point is that the definition of “number” and “infinity” depends on which branch of math you’re working in.
Yes, but in that case it’s the same amount. For every real x in the first interval there is a real y=2x in the second. Also for every real y in the second interval there is a real x=y/2 in the first.
That’s not true, those two infinities are functionally the same. There are bigger infinities than others, or at least higher orders, but those two are the same order.
The cardinal numbers have entered the chat. Along with the ordinal numbers, surreal numbers, extended real numbers, projective extended real numbers, wheels, and Riemenn sphere.
oo
oh, wrong thread
∞…
shush
∀R { { ∀[ψ], t: R([ψ],t) ↔ ([ψ] = “xi ∈ xj” ∧ t(xi) ∈ t(xj)) ∨ ([ψ] = “xi = xj” ∧ t(xi) = t(xj)) ∨ ([ψ] = “(¬θ)” ∧ ¬R([θ], t)) ∨ ([ψ] = “(θ∧ξ)” ∧ R([θ], t) ∧ R([ξ], t)) ∨ ([ψ] = “∃xi(θ)” ∧ ∃t′: R([θ], t′)) (where t′ is a copy of t with xi changed) } ⇒ R([ϕ],s) }
Ah, good ol’ Rayo’s number. I can fit that in a twitter post!
9!!!!!!!!!!![...]False! Create a new number system, base of whatever the chart will accept, then full post of that. :3
Just use an existing system FFFFFFFFFFFFFFFFFFFFFFFFFF
Why stop at base 16?
ZZZ[…]
Joke’s on you, the FFFFFFFFFFFFFFFFFFF was actually written in base 9!!!.. and F simply happened to be mapped to the last integer
But there’s still soooo much left on the tableeeeeeeeeeee!!!
TREE(3)
TREE(4)
tree(tree(tree(tree(tree(tree(tree(tree(tree(tree(tree(tree(3))))))))))))
BB(tree(3))
I’m unfamiliar, what is BB?
Busy beaver algorithm. https://wiki.bbchallenge.org/wiki/Busy_Beaver_Functions
Starting definition: the largest number of steps (or shifts) that any Turing machine (of a certain size, and starting with a blank tape) takes before halting.
Computerphile does a good treatment on it. https://www.youtube.com/watch?v=CE8UhcyJS0I
tree() is a different function to TREE()
TREE(fiddy)
Fuckin loch ness monster
Maybe grade 4 I said something rude or dumb and had to stay in while the whole class went on an adventure and I decided to determine the largest number ever and I wrote ‘9’ on my notebook about 50 times and then the book report and I never did learn, what is the largest ever number
John Madden! John Madden!AEIOU AEIOU AEIOU
replace 3 of the nines at the beginning with “10^”
I’m no mathematician, but surely replacing every three 9s with 10^ would be larger, right?
BB(9)
10 in base infinity I guess?
10 (base grahams number)
11 (base 10 (base Graham’s number))
I’m guessing “no operators” is implied.
But now I’m wondering if it would be worth it to sacrifice the two leading nines to add “0x” instead and replace all the other "9"s with "F"s
Haha have you found the 8?
Fuck I thought I fixed it
Huh, it’s qntm, the author of There Is No Antimemetics Division
