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## The Alpha Function

The Alpha Function uses Version 9 code and my S Function (substitution function) to generate every finite integer up to a very large number. The Alpha Function has a growth rate of $$f_{\varphi(1,1,0)}(n)$$ for any n.

The Alpha Function has one parameter: $$\alpha(r)$$ where r is any real number. The real number is manipulated to create a finite sequence of finite integers. The sequences can be translated to represent every unique finite integer.

Refer to the Granularity Examples in my Version 9 blog that illustrate how finely tuned the functions are to access any and every finite integer.

## What is the Alpha Function

My motivation to create this function was to develop a finely grained number notation system for really big numbers. $$\alpha(2)$$ for example can be used to reference the number 6. Therefore 2 is the Alpha Index for the number 6. Alpha needs to reference big numbers very quickly to be useful, therefore it uses The S Function for this purpose. Alpha should also be monotonically increasing and every input real $$a > b$$, results in a larger output number, where $$\alpha(a) >= \alpha(b)$$ in all cases. The function is finely grained. It accepts a real number input and depending upon the precision of the real number, it can locate any and every big number.

## High Level Description

The Alpha Function is 'calibrated' to accept real number inputs up to 100,000 and generate unique S() function outputs representing any and every big number.

## Some Calculations

Refer to my Version 9 blog blog for more examples:

$$\alpha(0.00) = S(0,0,0) = 0$$

$$\alpha(1.00) = S(2,1,1) = 4$$

$$\alpha(2.00) = S(S(2,1,1),0,2) = S(4,0,2) = 6$$

$$\alpha(3.00) = S(S(2,1,1),0,S(2,0,1)) = S(4,0,3) = 7$$

$$\alpha(4.00) = S(S(S(2,T(0),1),1,1),0,1) = S(S(8,1,1),0,1) = S(16,0,1) = 17$$

$$\alpha(4.29) = S(S(2,T(0),1),2,1) = f_2(f_{\omega}(2)) = f_2(8) = 2048$$

$$\alpha(4.332) = S(S(S(2,T(0),1),2,1),1,9) = f_1^9(2048) = 2048.2^9 = 1048576 >$$ One Million

$$\alpha(4.33681975) = S(S(8,2,1),1,S(S(8,1,5),0,S(S(8,1,3),0,2)))$$

$$= S(2048,1,S(8,1,5) + S(64,0,2)) = 2048.2^{256 + 66} = 17.10^{99} >$$ Googol

$$\alpha(4.35) = S(S(2,T(0),1),2,3) = S(8,2,3) = f_2^3(f_{\omega}(2)) = f_2^3(8) > f_2^3(6) >$$ Googolplex

$$\alpha(4.39) = S(S(2,T(0),1),3,1) = S(8,3,1)$$

$$\alpha(4.75) = S(2,T(0) + 1,1) = f_{\omega + 1}(2) = f_{\omega}(8)$$

$$\alpha(6.2596315) = S(S(2,S(T(0),0,1),1),T(0),S(S(8,1,2),0,S(16,0,S(8,0,7))))$$

$$= S(f_{\omega}(8),T(0),63) = f_{\omega}^{63}(f_{\omega}(8)) = f_{\omega}^{64}(8) > G$$ is Graham's number

$$\alpha(13.9) = S(2,S(T(1),0,1),1) = S(2,T(1),2) = f_{\varphi(\omega,0)}^2(2) > f_{\varphi(\omega,0)}(3)$$

$$\alpha(100) = S(S(2,S(S(T(T(0)),S(S(T(1),1,1),0,T(1)),1),0,S(S(T(1),S(T(0),0,1),T(0)),T(0),S(S(T(1),T(0),1),1,1))),1),S(T(T(0)),S(2,S(T(1),1,S(S(T(0),1,1),0,T(0))),1),1),1)$$

$$> S(2,T(T(0)),2) > f_{\varphi(1,0,0)}(3)$$

In general:

$$S(2,b+1,1) = S(2,b,2) = S(S(2,b,1),b,1) > S(n,b,1)$$ for any reasonably sized number $$n$$

Here are more comparisons:

$$\alpha(1306.35) = S(2,S(T(T(T(0))),0,1),1) > S(n,T^{3}(0),1) > f_{\varphi(1,1,0)}(3)$$

$$\alpha(9058.76) = S(2,S(T(T(T(T(0)))),0,1),1) > S(n,T^{4}(0),1) > f_{\varphi(1,1,0)}(4)$$

$$\alpha(100000) = S(2,T^{\omega}(0),1) = \omega$$ equals infinity by definition

## Further References

Further references to relevant blogs can be found here: User:B1mb0w