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Bigger and bigger

The fast growing hierarchy is a system to create huge numbers using ordinals. In order to create larger numbers, we need larger ordinals, so we can use the Veblen Hierarchy, which makes ordinals from numbers. Then we advance some more, and use ordinal collapsing functions to make large ordinals out of bigger ordinals. However, we have a hierarchy right there for us to make large numbers. Why not use it to create large ordinals?

Ruleset

The ruleset so far is the best I can do. If someone can think of a more formal ruleset, please tell me.

f_0(alpha) = alpha+1.

f_beta+1(alpha) = f_beta^(alpha)(alpha)

f_beta(alpha) = lim{f_beta[1](alpha), f_beta[2](alpha), f_beta[3](alpha)...} for all [n]<alpha

The last rule is applicable only when beta is a limit ordinal.

The second rule is all the problem. What does the "\(\omega\)^2"th member of a sequence mean, or the "\(\omega\)+1"th member, or even the \(\omega\)th member? Here we must identify the sequence and its limit, and apply it from there.

For example, take the sequence f_beta(n) = \(\omega\)*n. Its limit is \(\omega\)^2, and the limit is the \(\omega\)th member, as no finite number we place will reach \(\omega\)*\(\omega\), we need to use omega. However, we can reach higher using infinite ordinals. If we place for example, \(\omega\)^2, we reach \(\omega\)*\(\omega\)^2 = \(\omega\)^3. We can place something like w^w+1, and get w*(w^w+1) = w^(w+1)+w. With more complex functions we need to figure out the ordinals before we can generalize, and often it will be an approximation, but then, so it is with the regular fast growing hierarchy.

From f_0(\(\omega\)) = \(\omega\)+1 to f_\(\omega\)^\(\omega\)+1(\(\omega\)) = LVO

Here I have only defined f_alpha(w), as it is more simple to work with, and in this blog post only up to around f_w^2. I'll soon post more!

\begin{eqnarray*} f_0(\omega) &=& \omega + 1 \\ f_1(\omega) &=& f_0^\omega(\omega) = \text{lim} \lbrace \omega + 1, \omega + 2, \omega + 3 \cdots \rbrace = \omega2 \\ f_2(\omega) &=& f_1^\omega(\omega) = \text{lim} \lbrace \omega2, \omega4, \omega8 \cdots \rbrace = \omega2^\omega = \omega^2 \\ f_3(\omega) &=& f_2^\omega(\omega) = \text{lim} \lbrace \omega^2, \omega^4, \omega^8 \cdots \rbrace = \omega^{2^\omega} = \omega^\omega \\ f_4(\omega) &=& f_3^\omega(\omega) = \text{lim} \lbrace \omega^\omega, {(\omega^\omega)}^{(\omega^\omega)}, {({(\omega^\omega)}^{(\omega^\omega)})}^{({(\omega^\omega)}^{(\omega^\omega)})} \cdots \rbrace \approx \text{lim} \lbrace \omega^\omega, \omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}} \cdots \rbrace = \varepsilon_0 \\ f_5(\omega) &=& f_4^\omega(\omega) \approx \text{lim} \lbrace \varepsilon_0, \varepsilon_{\varepsilon_0}, \varepsilon_{\varepsilon_{\varepsilon_0}} \cdots \rbrace = \zeta_0 \\ f_{m+3}(\omega) &=& f_{m+2}^\omega(\omega) = \text{lim} \lbrace f_{m+2}(\omega), f_{m+2}(f_{m+2}(\omega)), f_{m+2}(f_{m+2}(f_{m+2}(\omega))) \cdots \rbrace \approx \varphi(m,0) \\ f_{\omega}(\omega) &=& \text{lim} \lbrace f_1(\omega), f_2(\omega), f_3(\omega) \cdots \rbrace \approx \text{lim} \lbrace \varphi(1,0), \varphi(2,0), \varphi(3,0) \cdots \rbrace = \varphi(\omega,0) \\ f_{\omega+1}(\omega) &=& f_{\omega}^\omega(\omega) \approx \text{lim} \lbrace \varphi(1,0), \varphi(\varphi(1,0),0), \varphi(\varphi(\varphi(1,0),0),0) \cdots \rbrace = \varphi(1,0,0) = \Gamma_0 \\ f_{\omega+2}(\omega) &\approx& \text{lim} \lbrace \Gamma_0, \Gamma_{\Gamma_0}, \Gamma_{\Gamma_{\Gamma_0}} \cdots \rbrace = \varphi(1,1,0) \\ f_{\omega+3}(\omega) &\approx& \text{lim} \lbrace \varphi(1,1,0), \varphi(1,1,\varphi(1,1,0)), \varphi(1,1,\varphi(1,1,\varphi(1,1,0))) \cdots \rbrace = \varphi(1,2,0) \\ f_{\omega+m+1}(\omega) &\approx& \varphi(1,\omega,0) \\ f_{\omega 2}(\omega) &=& \text{lim} \lbrace f_{\omega+1}(\omega), f_{\omega+2}(\omega), f_{\omega+3}(\omega) \cdots \rbrace \approx \text{lim} \lbrace \varphi(1,0,0), \varphi(1,1,0), \varphi (1,2,0) \cdots \rbrace \approx \varphi(1,\omega,0) \\ f_{\omega 2+1}(\omega) &\approx& \text{lim} \lbrace \varphi(1,\omega,0), \varphi(1,\varphi(1,\omega,0)), \varphi(1,\varphi(1,\varphi(1,\omega,0))) \cdots \rbrace = \varphi(2,0,0) \\ f_{\omega 3}(\omega) &\approx& \text{lim} \lbrace \varphi(2,0,0), \varphi(2,1,0), \varphi(2,2,0) \cdots \rbrace =(\varphi(2,\omega,0) \\ f_{\omega m}(\omega) &\approx& \varphi(m,\omega,0) \\ f_{\omega^2}(\omega) &=& \text{lim} \lbrace f_{\omega}(\omega), f_{\omega 2}(\omega), f_{\omega 3}(\omega) \cdots \rbrace \approx \text{lim} \lbrace \varphi(\omega,0), \varphi(1,\omega,0), \varphi(2,\omega,0) \cdots \rbrace \approx \varphi(\omega,\omega,0) \\ f_{\omega^2+1}(\omega) &=& f_{\omega^2}^\omega(\omega) \approx \text{lim} \lbrace \varphi(\omega,\omega,0), \varphi(\varphi(\omega,\omega,0),\omega,0), \varphi(\varphi(\varphi(\omega,\omega,0),\omega,0),\omega,0) \cdots \approx \varphi(1,0,0,0) \\ f_{\omega^3}(\omega) &\approx& \varphi(\omega,0,0,0) \\ f_{\omega^m}(\omega) &\approx& \varphi(\omega,\underbrace{0,0,0...}_m) \\ f_{\omega^\omega}(\omega) &=& \text{lim} \lbrace f_{\omega}(\omega), f_{\omega^2}(\omega), f_{\omega^3}(\omega) \cdots \rbrace \approx \text{lim} \lbrace \varphi(\omega,0), \varphi(\omega,0,0), \varphi(\omega,0,0,0) \cdots \rbrace \approx \vartheta(\Omega^\omega) \\ f_{\omega^\omega+1}(\omega) &=& f_{\omega^\omega}^\omega(\omega) \approx \text{lim} \lbrace \vartheta(\Omega^\omega), \vartheta(\Omega^{\vartheta(\Omega^\omega)}), \vartheta(\Omega^{\vartheta(\Omega^{\vartheta(\Omega^\omega)})}) \cdots \rbrace \approx \vartheta(\Omega^\Omega) \\ \end{eqnarray*}

Analysis of data

An interesting thing to see is that when alpha is a limit ordinal, f_alpha+1(w) is ridiculously more powerful than f_alpha(w). A good example is to see f_w(w) ~ phi(w,0) and f_w+1(w) ~ phi(1,0,0). This is because while f_w(w) is merely the limit of {f_1(w), f_2(w), f_3(w)...}, f_w+1(w) is repeated recursion on that limit, in the form of {f_w(w), f_w(f_w(w)), f_w(f_w(f_w(w)))...}, which is far, far greater. This explains how the SVO starts building up from f_w(w) = phi(w,0) all the way to f_w^w(w), but the LVO, a far, far greater ordinal appears immediately after, at f_w^w+1(w).

Future continuations

I will continue the fast growing hierarchy using f_alpha(w) up to at least w^3, and if I manage maybe even more, but I don't think I have a well enough understanding of the higher parts of ordinal collapsing functions. I'll try though! Furthermore, I may make Hardy hierarchy and slow growing hierarchy comparisons (though with slow growing hierarchy I obviously can't use omega: it would come out the same ordinal I put in!) and do generalizations for them (that is, specify a resulting ordinal for all ordinals, not only for omega).

Any questions, observations, corrections and comments are welcome!

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