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J Function Sandpit \(J_2\)

The J Function is a work in progress. This sandpit defines a function called \(J_2\) which contains ideas that will be used in the final J Function. Click here for the J Function blog.


Summary

The \(J_2\) function is a cleaner and more successful version of the \(J_0\) function. It uses one real number "r" as an input parameter. The function can then produce effectively every FGH function with ordinals up to \(\epsilon_0\) in order of increasing size, for any number n. This makes for a useful function for referencing some very large numbers which otherwise need complex notation that is difficult to read or understand.


Principles

Some principles used to create this function are:

Starting assumption: the function makes extensive use of the small integers 0 to 7. All other numbers created by the function will be some instance of the FGH function. Note that this starting range includes some trivial instances of FGH.

\(1 = f_0 (0)\)

\(2 = f_0 (1) = f_1 (1) = f_{\omega}(1) = f_{\epsilon_0}(1)\)

\(3 = f_0 (2)\)

\(4 = f_1 (2)\)

Generating FGH functions: the first FGH function is this trivial example.

\(8 = f_2(2) = f_{\omega}(2)\)

From here the function generates FGH functions in the following sequence:

\(f_2(2), f_2(3), f_2(4), f_2(5), f_2(6), f_2(7)\)

then

\(f_2^2(3), f_2^2(4), f_2^2(5), f_2^2(6), f_2^2(7)\)

note that only values of n = 2 to 7 are used. The sequence continues for higher nested powers "p" of the FGH function. When p = 2 then n = 2 is not used. It is an illegal entry because n > p in all cases. The sequence for p = 3 is shown next. This illustrates how the value of n is restricted by p.

\(f_2^3(3), f_2^3(4), f_2^3(5), f_2^3(6), f_2^3(7)\)

In this sequence \(f_2^3(3)\) should be illegal but because it can be represented differently as \(f_{\omega}(3)\) it is accepted in the sequence. For the next sequence of p = 4, it is clear that n = 3 is illegal but n = 4 is accepted.

\(f_2^4(4), f_2^4(5), f_2^4(6), f_2^4(7)\)

The reason n = 4 is accepted because it can be represented differently as \(f_3(4)\) and this illustrates the general rule for creating these ascending sequences. In this example the ordinal "w" of the FGH function is 3 and n > w and n > p at all times.

It is not difficult to see the sequence continuing to \(f_2^6 (7)\) after which it proceeds with:

\(f_3(4), f_3(5), f_3(6), f_3(7)\)

The ordinal w = 3 and the nested power p = 1. The value of n can then range from 4 to 7 following the same rules as before. The nested power can then be increased to p = 2 and following these rules we reach the FGH function \(f_6^7(7)\) which is accepted because it can be represented as \(f_{\omega}(7)\).

The sequence at this point has exhausted all the FGH functions using small numbers 0 to 7. We can now use the first generated instance of FGH function, namely \(f_2 (2)\), and replace n = 7 with this function.

\(f_{\omega}(f_2(2)) = f_{\omega}(f_{\omega}(2)) = f_{\omega}^2(2) = f_{\omega + 1}(2)\)

The sequence can continue at this point with n = 3 to 7 and then p = 2 to 7.

Work in progress


Definitions

The definition of the function is accurately defined in program code (to be provided). A high level definition is as follows:

\(J_2(n^n) = f_{(\omega\uparrow\uparrow(n-1))^{n+1}}(n+1)\)

This is equivalent to the following simpler definition:

\(J_2(n^n) = f_{\omega\uparrow\uparrow n}(n+1)\)

but the longer definition serves a useful purpose by forcing the sequence of numbers such as:

\(f_{\omega}(3) = f_3(3) = f_2^3(3)\) before \(f_2^3(4)\)

and

\(f_{\omega^{\omega}}(3) = f_{\omega^3}(3)\) before \(f_{\omega^3}(4)\)


Work in progress


Calculations

Here are some preliminary calculations for various values of r:

\(J_2(0) = 0\)

\(J_2(0.25) = 2\)

\(J_2(0.5) = 4\)

\(J_2(0.75) = 6\)

\(J_2(1) = 8\)

\(J_2(1.005) = f_2(3)\)

\(J_2(1.045) = f_2(7)\)

\(J_2(1.055) = f_2^2(3)\)

\(J_2(1.38) = f_3^2(6)\)

\(J_2(1.39) = f_3^2(7)\)

\(J_2(1.405) = f_3^3(4)\)

\(J_2(1.705) = f_4^3(5)\)

\(J_2(2.305) = f_6^3(7)\)

\(J_2(2.355) = f_6^4(7)\)

\(J_2(2.405) = f_6^5(7)\)

\(J_2(2.455) = f_6^6(7)\)

\(J_2(2.5) = f_{\omega}(7)\)

\(J_2(2.505) = f_{\omega}^2(6)\)

\(J_2(2.51) = f_{\omega}^4(6)\)

\(J_2(2.52) = f_{\omega+1}(6)\)

\(J_2(2.53) = f_{\omega+1}^5(6)\)

\(J_2(2.54) = f_{\omega+2}^2(5)\)

\(J_2(2.55) = f_{\omega+2}^5(7)\)

\(J_2(2.56) = f_{\omega+3}^3(4)\)

\(J_2(2.57) = f_{\omega+3}^6(7)\)

\(J_2(2.58) = f_{\omega+4}^3(7)\)

\(J_2(2.6) = f_{\omega+5}^4(7)\)

\(J_2(2.65) = f_{\omega.2+1}^3(4)\)

\(J_2(2.7) = f_{\omega.2+4}^2(5)\)

\(J_2(2.72) = f_{\omega.2+5}^2(7)\)

\(J_2(2.725) = f_{\omega.2+5}^4(7)\)

\(J_2(2.75) = f_{\omega.2+6}^6(7)\)

\(J_2(2.8) = f_{\omega.3+2}^5(7)\)

\(J_2(2.9) = f_{\omega.4+1}^3(6)\)

\(J_2(3) = f_{\omega.4+6}^6(7)\)

\(J_2(3.1) = f_{\omega.5+5}^4(7)\)

\(J_2(3.2) = f_{\omega.6+4}^2(7)\)

\(J_2(3.21) = f_{\omega.6+4}^5(7)\)

\(J_2(3.22) = f_{\omega.6+5}^2(7)\)

\(J_2(3.23) = f_{\omega.6+5}^6(7)\)

\(J_2(3.24) = f_{\omega.6+6}^3(7)\)

\(J_2(3.245) = f_{\omega.6+6}^5(7)\)

\(J_2(3.25) = f_{\omega^2}(2)\)

\(J_2(3.252) = f_{\omega^2}(3)\)

\(J_2(3.255) = f_{\omega^2}(4)\)

\(J_2(3.26) = f_{\omega^2}^2(2)\)

\(J_2(3.27) = f_{\omega^2}^3(4)\)

\(J_2(3.28) = f_{\omega^2}^4(5)\)

\(J_2(3.29) = f_{\omega^2}^5(6)\)

\(J_2(3.3) = f_{\omega^2}^6(7)\)

\(J_2(3.32) = f_{\omega^2+1}^2(6)\)

\(J_2(3.34) = f_{\omega^2+1}^5(6)\)

\(J_2(3.36) = f_{\omega^2+2}(3)\)

\(J_2(3.38) = f_{\omega^2+2}(6)\)

\(J_2(3.4) = f_{\omega^2+2}^2(5)\)

\(J_2(3.45) = f_{\omega^2+3}^5(6)\)

\(J_2(3.5) = f_{\omega^2+4}^4(6)\)

\(J_2(3.55) = f_{\omega^2+5}^4(6)\)

\(J_2(3.6) = f_{\omega^2+6}^4(7)\)

\(J_2(3.65) = f_{\omega^2.2}^2(3)\)

\(J_2(3.7) = f_{\omega^2.2}^4(6)\)

\(J_2(3.75) = f_{\omega^2.2}^6(7)\)

\(J_2(3.8) = f_{\omega^2.2+1}^3(4)\)

\(J_2(3.85) = f_{\omega^2.2+1}^5(6)\)

\(J_2(3.9) = f_{\omega^2.2+2}(4)\)

\(J_2(3.95) = f_{\omega^2.2+2}^2(3)\)

\(J_2(4) = f_{\omega^2.2+2}^3(3)\)


Work in progress

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