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## No Longer RelevantEdit

This blog is no longer relevant to my work on the S Function and Alpha Function. Please keep this in mind if you want to review this work.

## Comparing T() Functions to OrdinalsEdit

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 CountEdit

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.

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:

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

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

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

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

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

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

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

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

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

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

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

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

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