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Comparing T() Functions to Ordinals

The S Function grows quickly. Comparing \(T()\) functions, that are used recursively in the S Function definition, directly to ordinals should make it easier to compare the growth rate of the S Function.


Omega Count

Using the Omega Count function of counting the number of \(\omega\) additions that are required to reach larger ordinals is one way of comparing \(T()\) functions to ordinals.

We can start with this comparison:

Let \(n=3\) and diagonalise over \(n\)

\(\Omega(n) = \omega == T(0) = n\)

\(\Omega(n.2) = \omega.2 == S(T(0),1,1) = n.2\)

\(\Omega(n^n) = \omega^{\omega} == S(S(T(0),2,1),0,T(0)) = n.2^n + n = n^n\) when \(n = 3\)

Here is a comprehensive comparison of \(T()\) functions to ordinals for \(n=2\):

\(\Omega(n) = \omega == T(0) = n\)

\(\Omega(n.2) = \omega.2 = \omega^{\omega} = \epsilon_0 == S(T(0),1,1) = n.2^1 = n.2\)

\(\Omega(n.2.2) = \epsilon_0.\omega == T(1) = n.2^n = 8\)

\(\Omega(4^4) = \epsilon_0^{\epsilon_0} = \epsilon_1\)

\(== S(T(1),1,S(S(T(0),1,1),0,1)) = 8.2^{2.2^1 + 1} = 2^3.2^5 = 2^8 = 256 = 4^4\)

\(\Omega(256^{256}) = \epsilon_1^{\epsilon_1} = \epsilon_{\omega}\)

\(<< S(T(1),T(0),T(0)) = 8.2^8.2^{8.2^8} = 8.256.2^{8.256} = 8.256.(2^8)^{256} = 8.256.256^{256} = 8.256^{257}\)

From here we can use the equality \(S(x,2,2) >> x\uparrow\uparrow 2\) to access higher ordinals:

\(\epsilon_{\omega + 1} << S(S(T(1),T(0),T(0)),2,2) = S(S(T(1),T(0),T(0)),T(0),T(0)) = S(T(1),T(0),S(T(0),1,1))\)

\(\epsilon_{\omega + 2} = \epsilon_{\epsilon_0} = \varphi(\omega,0) << S(T(1),T(0),S(T(0),1,2)) = S(T(1),T(0),T(1)) = S(T(1),S(T(0),0,1),1)\)

\(\varphi(1,\varphi(\omega,0)+1) << S(S(T(1),S(T(0),0,1),1),2,2) = S(S(T(1),S(T(0),0,1),1),T(0),T(0))\)

\(\varphi(\omega,1) << S(S(T(1),S(T(0),0,1),1),T(0),S(T(0),1,1))\)

\(\varphi(1,\varphi(\omega,1)+1) << S(S(T(1),S(T(0),0,1),1),T(0),S(T(0),1,2)) = S(S(T(1),S(T(0),0,1),1),T(0),T(1))\)

\(\varphi(\omega,\omega) << S(S(T(1),S(T(0),0,1),1),T(0),S(T(1),1,1))\)

\(\varphi(\omega,\omega + 1) << S(S(T(1),S(T(0),0,1),1),T(0),S(T(1),1,T(0)))\)

\(\varphi(\omega,\varphi(1,0)) << S(S(T(1),S(T(0),0,1),1),T(0),S(T(1),1,S(T(0),0,1)))\)

\(\varphi(\omega,\varphi(1,0) + 1) << S(S(T(1),S(T(0),0,1),1),T(0),S(T(1),1,S(T(0),1,1)))\)

\(\varphi(\omega,\varphi(1,0) + \omega + 1) << S(S(T(1),S(T(0),0,1),1),T(0),S(T(1),T(0),1)))\)

\(\varphi(\omega,\varphi(1,0).\omega) << S(S(T(1),S(T(0),0,1),1),T(0),S(S(T(1),T(0),1),0,1)))\)

\(\varphi(\omega,\varphi(1,0).(\omega + 1)) << S(S(T(1),S(T(0),0,1),1),T(0),S(S(T(1),T(0),1),0,S(S(T(0),1,1),0,1))))\)

\(\varphi(\omega,\varphi(1,0)^{\omega}) << S(S(T(1),S(T(0),0,1),1),T(0),S(S(T(1),T(0),1),0,S(T(1),0,1))))\)

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