In memory of a great mathematician …

John Horton Conway.

This will be Conway’s Chained Arrow Notation, created by Conway and R. K. Guy.

If you read this and you suddenly see CAN, that means Chained Arrow Notation.

CAN is defined as follows:

• a → b = a^b
• a → b → c = a ↑^c b = a ↑↑↑↑↑ b with c ↑’s
• a → … → b → 1 = a → … → b (when the last entry is 1, it is ignored.)
• a → … → b → 1 → c = a → … → b
• a → … → b → (c + 1) → (d + 1) = a → … → b → (a → … → b → c → (d + 1)) → d

→ is not an “operation” in the state that a → b → c is equal to neither (a → b) → c nor a → (b → c).

A function named cg(n), after Conway and Guy, starts with the following: 1, 2 → 2, 3 → 3 → 3, 4 → 4 → 4 → 4, …

It starts as 1, 4, 3^{7625597484987}, and 4 → 4 → 4 → 4 > {4, 4, 3, 2}, using Bird’s Proof. That means cg(4) > Graham’s Number. In FGH, it is about as fast as f_ω²(n). In Notation Array Notation, it can be represented as (n{3,n}n). What?

The two extensions are by Peter Hurford and Cookiefonster, respectively. Cookiefonster is a user of the Googology Wikia.

First, Peter Hurford’s extension!

a →_c b = a →_c-1 a →_c-1 a →_c-1 a →_c-1 a →_c-1 … →_c-1 a, with b →_c-1‘s. The basic rules are the same as the normal. The normal and extended versions can not be mixed up. Hurford showed that f(n) = n →_n n is about f_ω³(n) in the FGH.

He defined C(n) as follows:

• C(a) = a →_a a
• C(a, 1) = a →_C(a) a
• C(a, b) = a →_{C(a, b-1)} a
• C(a, 1, 1) = C(a, C(a, a))
• C(a, b, 1) = C(a, C(a, b-1, 1))
• C(a, 1, c) = C(a, C(a, a, c-1), c-1)
• C(a, b, c) = C(a, C(a, b-1, c), c-1)

f(n) = C(n, n, n) grows about as fast as f_ω³+ω(n) in the FGH.

Second, Cookiefonster’s extension!

Cookiefonster used superscripts for this.

Rules!

• a →¹ b = a^b
• a →^x b = a →^x-1 a →^x-1 a →^x-1 a →^x-1 … →^x-1 a, with b copies of a
• # →^x 1 = #, where # is an arbitrarily long part of the chain before the relevant terms
• # →^x 1 →^y a = #
• The last two rules are here.

f(n) = n →ⁿ n is about as fast growing as f_ω^ω(n) in the FGH.

Sorry for not being able to upload much. I had a lot of school stuff and couldn’t work on my blog. Sorry!

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