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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”
∞ ↑↑↑↑↑… ↑↑↑∞
Infinty is a magnitude, not a number
Also BB(8000) can (proven ) not be represented by ZFC so that might take the cake
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
∀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!
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
∞
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.
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.
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
“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 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.
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
10 (base grahams number)
11 (base 10 (base Graham’s number))
