User blog:Dchew89/Possible (Uncomputable) Function with Growth Rate of w 1^CK

The sequence described by this function starts out with {0,1,...}. From there, the overall function is described as follows: "based on all natural numbers less than or equal to the largest previous number in the sequence, a new sub-function must be defined so that it is more powerful than all functions previously defined, so that at least one input value less than the largest number in the sequence and greater than or equal to zero does not output a value larger than the largest previous number, and at the same time so that the same input value outputs a number larger than all previous functions defined, using the same input for each. If a function is definable given these conditions, and if it is larger than another definable function, it must output a number larger than the other definable function with at least one given input in a valid test to become defined in its place. If two definable functions of different strengths are equivalent given the same maximum input value, the more powerful of the two functions becomes defined, but only if other numerical values used to in the definition of the function are less than or equal to those in the other functions, and if no constants are present, the most significant instance of the input may be considered in the place of a constant. Any function defined must be strictly increasing for natural number inputs, and must be able to output a number larger than the previous largest number given that number as an input. The newly defined most powerful function will then accept the previous number in the sequence as its input, and the next number in the sequence will be its output. The process is then repeated."

To give an example, the sequence begins with no sub-functions described, so the first function defined would need to be $$n+1$$, as this is the only function that can act on either $$0$$ or $$1$$ and output a value between $$0$$ and $$1$$ inclusive, while at the same time, a more powerful function such as $$10^n$$ would not be defined because they do not output numbers larger than $$1$$ for the given inputs, and $$n+2$$ would output numbers larger than $$1$$ for any possible inputs. Therefore the sequence is now {0,1,2,...}, and the next function could only be $$n+2$$ because other options such as $$n*2$$, $$n^2$$, etc would give the same or smaller outputs for any possible input, despite being stronger. Now {0,1,2,4,...},$$n*2$$ still cannot be defined because it only overpowers $$n+4$$ when $$n=4$$, but inputting $$4$$ for either of those two functions would output a number $$>4$$, making that test invalid, and making $$n+4$$ the new strongest function. Now {0,1,2,4,8,...}, more powerful functions can finally be described, $$n^3$$ outgrowing $$n+8$$, $$n*4$$, etc, but functions like $$2^n$$ would not work given the same input, either growing too fast or too slow. Now {0,1,2,4,8,512,...}, a wide variety of different possibilities now appear, one particularly powerful one being $$n\uparrow\uparrow3$$, which, whether or not the most powerful possibility, would already output $$512^{512^{512}}$$, and contained in that, $$n\uparrow\uparrow\uparrow3$$. Eventually, there comes a point where $$n \uparrow^n 3$$ and other much more powerful functions come into play, as given very small outputs, as long as the largest sequence value is large enough, they can give outputs larger than their competition, for example, $$n\uparrow\uparrow\uparrow\uparrow6$$ may be a possible new function while inputting $$4$$ for $$n$$, but if $$n\uparrow\uparrow\uparrow\uparrow\uparrow6$$ is also a possible function when inputting $$5$$ for $$n$$, $$n \uparrow^n 6$$ could be chosen instead. For clarification, if, for example, the choice is between $$n\uparrow\uparrow\uparrow{n}$$ and $$n\uparrow\uparrow\uparrow3$$, and $$3$$ is the largest input to both that outputs smaller than the current largest sequence number, then $$n\uparrow\uparrow\uparrow{n}$$ must be chosen. Same goes for $$3\uparrow\uparrow\uparrow{n}$$, which is chosen, vs $$n\uparrow\uparrow\uparrow3$$, with a maximum input of $$3$$, as well as $$n ↑^n n$$, which is chosen, vs $$n\uparrow\uparrow\uparrow\uparrow{n}$$, with a maximum input of $$4$$.

Sequence of LOWER BOUNDS adapted to the FGH:

$$d(0) = 0$$

$$d(1) = 1$$

$$d(2) = f_0(1) = 2$$

$$d(3) = f^2_0(2) = 4$$

$$d(4) = f_1(4) = f_0(f_0(2)) = 8$$

$$d(5) = f_2(8) = f^2_2(2) = 2048$$

$$d(6) = f_3(2048) = f_3(f^2_2(2)) = f^{2050}_2(2)$$

$$d(7) >= f_n(d(6)) = f_{\omega}(d(6)) > f^2_{\omega}(2) = f_{\omega+1}(2)$$

$$d(8) >= f_{\omega+1}(d(7)) > f_{\omega+2}(2)$$

$$d(9) >= f_{\omega+2}(d(8)) > f_{\omega+3}(3) > f_{\omega+2}(3) > f_{\omega+2}(2)$$

$$f_{\omega+2}(2) = f_{\omega2}(2) = f_{\omega^2}(2) = f_{\omega^\omega}(2) = f_{\epsilon_0}(2)$$

$$d(10) >= f_{\epsilon_0}(d(9)) > f^2_{\epsilon_0}(2)$$

$$d(11) >= f_{\epsilon_0+1}(d(10))$$

$$d(12) >= f_{\epsilon_0+2}(d(11))$$

$$d(13) >= f_{\epsilon_0+\omega}(d(12))$$

It appears that as long as there is an ordinal in the FGH with a fundamental sequence built recursively from $$\omega$$, this function can reach it given a finite input; The function turns any $$2$$ in any possible FGH sub-function into an $$n$$, or an $$\omega$$ at either when the $$2$$ appears, or after the next output, and certainly eventually. It, however, should never reach the fundamental sequence of $$\omega_1^\text{CK}$$, as another ordinal can always be defined with a smaller growth rate, making $$\omega_1^\text{CK}$$ the apparent limit.