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So Graham's number or ##(10^{80})!##

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- Thread starter MathJakob
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- #1

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So Graham's number or ##(10^{80})!##

- #2

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which, if I am not mistaken, is far less than [itex]g_1[/itex].

- #3

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So ##(10^{80})!## isn't even as large as g1... and Graham's number is g64... WOW

- #4

arildno

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Arildno number =Graham's number+1.

Can I also get a WOW! from you?

- #5

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Arildno number =Graham's number+1.

Can I also get a WOW! from you?

No. Your number is meaningless, it's never been used in any meaningful way.

- #6

arildno

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No. Your number is meaningless, it's never been used in any meaningful way.

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an extremely weak lower bound for TREE(3), is A(A(...A(1)...)), where the number of As is A(187196)

Graham's number, for example, is approximately A^64(4) which is much smaller than the lower bound A^(A(187196))(1).

A is the Ackermann function.

http://en.wikipedia.org/wiki/Kruskal's_tree_theorem

- #8

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So Graham's number or ##(10^{80})!##

Tip: if you can write out the number in any "normal" way, it's waaaaaaaaaaaaaaaaay smaller than Graham's number.

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- #10

arildno

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It seems the Arildno number is a dwarf, after all..

- #11

Mentallic

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It seems the Arildno number is a dwarf, after all..

Poor thing, the Arildno number is coming under lots of fire in here!

I haven't really been able to find any sources that explain some of the larger operators in the fast growing hierarchy. For example,

http://googology.wikia.com/wiki/Fast-growing_hierarchy

gives a list of growth rates, but there's a lot of details being left out. Sadly, I only understand how to calculate up to and including [itex]f_{\epsilon_0}(n)[/itex].

And to do so, I use a calculator of course

- #12

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Beyond [itex]\varepsilon_0[/itex] we have:

[itex]\varepsilon_1 = \varepsilon_0 ^ {\varepsilon_0 ^ {\varepsilon_0 ^\ddots}}[/itex]

[itex]\varepsilon_2 = \varepsilon_1 ^ {\varepsilon_1 ^ {\varepsilon_1 ^\ddots}}[/itex]

[itex]\varepsilon_\omega = \sup \lbrace \varepsilon_0, \varepsilon_1, \varepsilon_2, \ldots \rbrace [/itex]

and so on. Eventually we get to

[itex]\zeta_0 = \varepsilon_{\varepsilon_{\varepsilon_\cdots}}[/itex]

We can set [itex]\varphi(0, \alpha) = \omega^\alpha, \varphi(1, \alpha) = \varepsilon_\alpha, \varphi(2, \alpha) = \zeta_\alpha[/itex]. More generally,

[itex]\varphi(\alpha+1, \beta) = [/itex] the [itex]\beta[/itex]th fixed point of [itex]f(\gamma) = \varphi(\alpha, \gamma)[/itex].

When [itex]\alpha[/itex] is a limit ordinal, [itex]\varphi(\alpha, \beta)[/itex] is the [itex]\beta[/itex]th ordinal in the intersection in the ranges of [itex]f(\delta) = \varphi(\gamma, \delta)[/itex] for all [itex]\gamma < \alpha[/itex].

So, for example,

[itex]\varphi(3, 0) = \varphi(2, \varphi(2, \varphi(2, \ldots)))[/itex]

[itex]\varphi(\omega, 0) = \sup (\varphi(1, 0), \varphi (2, 0), \varphi (3, 0), \ldots)[/itex]

and so on. This takes us up to

[itex]\Gamma_0 = \varphi(\varphi(\varphi(\ldots, 0),0),0)[/itex]

or alternatively, [itex]\Gamma_0[/itex] is the smallest ordinal [itex]\alpha[/itex] such that [itex]\alpha = \varphi(\alpha, 0)[/itex].

We can continue the notation with [itex]\varphi(1, 0, 0) = \Gamma_0[/itex], and [itex]\varphi(1, 0, \alpha)[/itex] is the [itex]\alpha[/itex]th fixed point of [itex]f(\beta) = \varphi(\beta, 0)[/itex].

The ordinals [itex]\varphi (1, \alpha, \beta)[/itex] are defined analogously to [itex]\varphi(\alpha, \beta)[/itex], i.e. each function [itex]f(\beta) = \varphi (1, \alpha+1, \beta)[/itex] enumerates the fixed points of [itex]g(\beta) = \varphi(1, \alpha, \beta)[/itex], and at limit ordinals you enumerate the intersection of the ranges of previous ordinals. Then [itex]\varphi(2, 0, \alpha)[/itex] is the [itex]\alpha[/itex]th fixed point of [itex]f(\beta) = \varphi(1, \beta, 0)[/itex], and you can construct the hierarchy [itex]\varphi(2, \alpha, \beta)[/itex] similarly. At limit ordinals you take the intersection of the ranges again, and so we define [itex]\varphi (\alpha, \beta, \gamma)[/itex] for all ordinals [itex]\alpha, \beta, \gamma[/itex]. Then we can define

[itex]\varphi (1, 0, 0, \alpha)[/itex] to be the [itex]\alpha[/itex]th fixed point of [itex]f(\beta) = \varphi(\beta, 0, 0)[/itex].

We can then define [itex]\varphi(\alpha, \beta, \gamma, \delta), \varphi(\alpha, \beta, \gamma, \delta, \epsilon)[/itex], and so on. The general definition for the n-ary Veblen funciton is:

[itex]\varphi (\alpha) = \omega^{\alpha}[/itex]

[itex]\varphi (\alpha_1, \alpha_2, \ldots, \alpha_n + 1, 0, \ldots, 0, \beta)[/itex] is the [itex]\beta[/itex]th fixed point of the function [itex]f(\gamma) = \varphi(\alpha_1, \alpha_2, \ldots, \alpha_n, \gamma, 0, \ldots, 0)[/itex]

When [itex]\alpha_n[/itex] is a limit ordinal, [itex]\varphi (\alpha_1, \alpha_2, \ldots, \alpha_n, 0, \ldots, 0, \beta)[/itex] is the [itex]\beta[/itex]th ordinal in the intersection of the ranges of [itex]f(\delta) = \varphi(\alpha_1, \alpha_2, \ldots, \alpha_{n-1}, \gamma, \delta, 0, \ldots, 0)[/itex] for all [itex]\gamma < \alpha_n[/itex].

We define the Small Veblen Ordinal as

[itex]\sup (\varphi (1, 0), \varphi(1, 0, 0), \varphi(1, 0, 0), \ldots)[/itex].

TREE(n) is larger than the fast-growing hierarchy at the level of the Small Veblen Ordinal. (how much further is not known.).

I'll go one step further: we can extend the n-ary Veblen function to transfinitely many places. Obviously, we can't write out transfinitely many variables, so we need to modify our notation: instead of

[itex]\varphi(\alpha, \beta, \gamma, \delta, \epsilon)[/itex]

we write

[itex]\varphi(\alpha @ 4, \beta @ 3, \gamma @ 2, \delta @ 1, \epsilon @ 0)[/itex].

So we append "@ n" to every variable, where n represents the index of the variable. This allows us to skip variables that are 0, and so we can notate things like [itex]\varphi (1 @ \omega, \alpha @ 0)[/itex]. [itex]\varphi(1 @ \omega, \alpha @ 0)[/itex] is defined as the [itex]\alpha[/itex]th ordinal that is a fixed point of [itex]f(\beta) = \varphi (\beta @ n)[/itex] for all [itex]n < \omega[/itex].

More generally, we define

[itex]\varphi(\alpha_1 @ \beta_1, \alpha_2 @ \beta_2, \ldots, \alpha_n + 1 @ \beta_n + 1, \gamma @ 0)[/itex] is the [itex]\gamma[/itex]th fixed point of [itex]f(\delta) = \varphi(\alpha_1 @ \beta_1, \alpha_2 @ \beta_2, \ldots, \alpha_n @ \beta_n + 1, \delta @ \beta_n)[/itex]

When [itex]\alpha_n[/itex] is a limit ordinal, [itex]\varphi(\alpha_1 @ \beta_1, \alpha_2 @ \beta_2, \ldots, \alpha_n @ \beta_n + 1, \gamma @ 0)[/itex] is the [itex]\gamma[/itex]th ordinal in the intersection of the ranges of [itex]f(\epsilon) = \varphi(\alpha_1 @ \beta_1, \alpha_2 @ \beta_2, \ldots, \delta @ \beta_n + 1, \epsilon @ \beta_n)[/itex] for all [itex]\delta < \alpha_n[/itex]

When [itex]\beta_n[/itex] is a limit ordinal, [itex]\varphi(\alpha_1 @ \beta_1, \alpha_2 @ \beta_2, \ldots, \alpha_n + 1 @ \beta_n, \gamma @ 0)[/itex] is the [itex]\gamma[/itex]th ordinal in the intersection of the ranges of [itex]f(\epsilon) = \varphi(\alpha_1 @ \beta_1, \alpha_2 @ \beta_2, \ldots, \alpha_n @ \beta_n, \epsilon @ \delta)[/itex] for all [itex]\delta < \beta_n[/itex]

When [itex]\alpha_n[/itex] and [itex]\beta_n[/itex] are limit ordinals, [itex]\varphi(\alpha_1 @ \beta_1, \alpha_2 @ \beta_2, \ldots, \alpha_n @ \beta_n, \gamma @ 0[/itex] is the [itex]\gamma[/itex]th ordinal in the intersection of the ranges of [itex]f(\zeta) = \varphi(\alpha_1 @ \beta_1, \alpha_2 @ \beta_2, \ldots, \delta @ \beta_n, \zeta @ \epsilon)[/itex] for all [itex]\delta < \alpha_n, \epsilon < \beta_n[/itex]

This defines [itex]\varphi(\alpha_1 @ \beta_1, \ldots, \alpha_n @ \beta_n)[/itex] for all [itex]\alpha_i[/itex] and [itex]\beta_i[/itex]. This notation is known as Schutte's Klammersymbolen.

The smallest ordinal [itex]\alpha[/itex] such that [itex]\alpha = \varphi (1 @ \alpha)[/itex] is known as the Large Veblen Ordinal. I would think that TREE(n) would not reach the Large Veblen Ordinal in the fast-growing hierarchy, but I don't think this is known.

Phew! I hope this was at least somewhat comprehensible.

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