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The SeT function offers a way to generate very large string sequences (representing large numbers). It is the third version of my substitution functions and it has a growth rate \(\approx f_{LVO}(n)\).
The SeT function is based on my other substitution function called The S Function (Version 2). Refer to my other blogs for more information on my work.
The SeT function is recursively defined set of two functions \(S()\) and \(T()\) which use string substitution procedures only. They do not explicitly use any mathematical or transfinite ordinal theory. In the SeT function, the \(S()\) function behaves the same as it does in The S Function (Version 2), but the \(T()\) function has been extended. The new substitution function is called S and ext…
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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.
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.
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 \(n\)
\(\Omega(n) = \omega == T(0) = n\)
\(\Omega(n.2) = \omega.2 == S(T(0),1,1) = n.2\)
\(\Omega(n^n) = \omega^{\omega} == S(S(T(0),2,1),0,T(0)) = n.2^n + n = n^n\) wh…
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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.
I think it is useful to keep track of the number of \(\omega\) additions are required to reach higher ordinals. Here is an example of what I mean:
Let \(n = 2\) and diagonalise over \(n\) then:
\(\omega + \omega = \omega.2 = \omega^2 = \epsilon_0 = \varphi(1,0)\)
Now lets define an Omega Count function that keeps track of this count:
\(\Omega(2) = \omega^2 = \epsilon_0 = \varphi(1,0)\)
\(\Omega(2^2) = \Omega(4) = \epsilon_0^{\epsilon_0} = \epsilon_1 = \varphi(1,1)\)
\(\Omega(4^4) = \epsilon_1^{\epsilon_1} = \epsilon_2 = \varphi(1,2) = \varphi(1,\omega)\)
Here are the Omega Counts for various ordinals and vario…
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This blog will provide a detailed calculation and references for the growth rate of The S Function that I have developed. The growth rate is comparable to:
\(S(n,T^{T(0)}(0),1) \approx f_{\varphi(1,1,0)}(n)\)
The S function is recursively defined set of two functions \(S()\) and \(T()\) which use string substitution procedures only. S Functions can be either restricted or generalised. Refer to my main blog on The S Function for a full definition of how the function is constructed.
As a simple introduction it will be useful to compare a typical S function with more familiar functions:
\(S(3,2,1) = f_2(3)\) ordinal value
This equivalence is intentional. In fact:
\(S(n,g,h) = f_g^h(n)\) ordinal value
This equivalence will become more obvious if you ref…
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This is my candidate for a large named number. It is called The Last Alpha Number and is calculated using The Alpha Function. It is approximately \(f_{svo}(10000)\) and therefore it is not big (by this website standard).
The Alpha Function calculates an ordinal for a given real number input. For example:
\(\alpha(0.00) = 0\)
\(\alpha(1.00) = 1\)
\(\alpha(2.00) = 8\)
\(\alpha(e) = 9\)
\(\alpha(3.00) = 10\)
\(\alpha(\pi) = 11\)
\(\alpha(4.00) = f_{\omega}(8)\)
The function accepts real numbers up to 10,000 at which point it asymptotically goes to infinity. The Alpha Function is based on my work on the S Function (substitution function). The limit of the Alpha Function is represented as follows:
\(\alpha(10000) = S(2,T^{\omega}(0),1) = \omega\) equals in…
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