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## The SeT Function (substitution function) Version 3

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.

## What are SeT Function strings

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 extended T function or SeT for short.

The growth rate of the S Function is $$f_{svo}(n)$$ and the SeT Function grows at a rate beyond $$f_{LVO}(n)$$.

The S function is defined recursively as follows:

$$S(a,b,c)$$ where $$a,b,c$$ can be finite integer $$n$$, an $$S()$$ function or a $$T()$$ function

$$S(a,b,0) = a$$ for any $$a$$ or $$b$$

The extended T function is defined recursively as follows:

$$T(p_1;p_2;p_3; ... p_i)$$ where $$p$$ is a partition of finitely many parameters $$(d_1,d_2, ... d_j)$$.

Each parameter $$d$$ can be a finite integer $$n$$, an $$S()$$ function or a $$T()$$ function

## Definition of SeT Function strings

In the SeT function, the $$S()$$ function behaves the same as it does in The S Function (Version 2). Refer to that blog for more detail.

$$S()$$ function strings can be equivalent (=) or in an ascending order (<). This is evaluated between arbitrary S function strings using two string substitution procedures:

$$sub()$$

$$dec()$$

Refer to my other blog The S Function (Version 2) for more detail.

## The extended T Function

The T Function is extended to support multiple parameters and multiple partitions (of multiple parameters). The ruleset for the T Function is:

$$T(1) = S(T(0),T(0),1)$$

or

$$T(d) = S(T(dec(d)),S(dec(d)),1)$$

The ruleset for the extended T Function with multiple parameters is:

$$T(1,0) = T^{T(0)}(0)$$

$$T(1,m + 1) = T^{T(1,m)}(T(1,m))$$

$$T(2,0) = T^{T(1,0)}(1,0_*)$$

$$T(2,m + 1) = T^{T(2,m)}(1,0_*)$$

$$T(k + 1,0) = T^{T(k,0)}(k,0_*)$$

$$T(k + 1,m + 1) = T^{T(k + 1,m)}(k,0_*)$$

$$T(1,0,0) = T^{T(1,0)}(T(0)_*,T(0))$$

$$T(1,0,1) = T^{T(1,0,0)}(T(0)_*,T(0))$$

$$T(1,1,0) = T^{T(1,0,0)}(1,0,0_*)$$

$$T(2,0,0) = T^{T(1,0,0)}(1,0_*,0)$$

$$T(2,0,1) = T^{T(2,0,0)}(1,0_*,0)$$

$$T(2,1,0) = T^{T(2,0,0)}(2,0,0_*)$$

The ruleset for the extended T Function with multiple partitions of parameters is:

$$T(0;0) = T(0)$$

$$T(1;0) = T(0;1,0_{[T(1,0)]}) = T(1,0_{[T(1,0)]})$$

$$T(1;m + 1) = T(1,0_{[T(1,m)]})$$

$$T(1;1,0) = T^{T(1;0)}(1;0_*)$$

$$T(1;1,1) = T^{T(1;1,0)}(1;0_*)$$

$$T(1;2,0) = T^{T(1;1,0)}(1;1,0_*)$$

$$T(2;0) = T(1;1,0_{[T(1,0)]})$$

$$T(j + 1;0) = T(j;1,0_{[T(j,0)]})$$

$$T(1;0;0) = T(1,0_{[T(1,0)]};0)$$

## Some Example SeT Function strings

Here are some example SeT Function strings in ascending order:

$$0$$

$$1$$

$$2$$

$$3$$

$$S(3,0,1)$$

$$S(3,0,2)$$

$$S(3,1,1)$$

$$S(S(3,1,1),0,1)$$

$$S(S(3,1,1),0,2)$$

$$S(S(3,1,1),0,S(3,0,1))$$

$$S(3,1,2)$$

$$S(3,2,1)$$

$$S(3,T(0),1)$$

$$S(3,T(0),2)$$

$$S(3,S(T(0),0,1),1)$$

$$S(3,S(T(0),0,1),2)$$

$$S(3,S(T(0),0,2),1)$$

$$S(3,S(T(0),1,1),1)$$

$$S(3,S(T(0),2,1),1)$$

$$S(3,T(1),1)$$

$$S(3,S(T(1),0,T(0)),1)$$

$$S(3,S(T(1),T(0),0),1)$$

$$S(3,T(2),1)$$

$$S(3,T^2(0),1)$$

$$S(3,T(1,0),1)$$

$$S(3,T(1,0,0),1)$$

$$S(3,T(1,0_{[T(0)]}),1)$$

$$S(3,T(1,0_{[T(1)]}),1)$$

$$S(3,T(1,0_{[T(2)]}),1)$$

$$S(3,T(1,0_{[T^2(0)]}),1)$$

$$S(3,T(1;0),1)$$

$$S(3,T(1;1),1)$$

$$S(3,T(1;T(0)),1)$$

$$S(3,T(1;T^2(0)),1)$$

$$S(3,T(1;1,0),1)$$

$$S(3,T(1;1,0,0),1)$$

$$S(3,T(1;1,0_{[T(0)]}),1)$$

$$S(3,T(1;1,0_{[T(1)]}),1)$$

$$S(3,T(1;1,0_{[T(2)]}),1)$$

$$S(3,T(1;1,0_{[T^2(0)]}),1)$$

$$S(3,T(2;0),1)$$

$$S(3,T(T(0);0),1)$$

$$S(3,T(T^2(0);0),1)$$

$$S(3,T(1,0;0),1)$$

$$S(3,T(1,0,0;0),1)$$

$$S(3,T(1;0;0),1) = S(3,T(1,0_{[T(1,0)]};0),1) = S(3,T(1,0_{[T^3(0)]};0),1)$$

## Growth Rate of the SeT Function

The number of SeT Function strings that has a growth rate beyond $$f_{LVO}(n)$$. Here are some example calculations:

$$S(n,T(0),1) = f_{\omega}(n)$$

$$S(n,S(T(0),3,1),1) > f_{\varphi(1,0)}(n) = f_{\epsilon_0}(n)$$

$$S(n,T(1),1) \approx f_{\varphi(\omega,0)}(n)$$

$$S(n,T(2),1) \approx f_{\varphi^2(1_*,0)}(n) = f_{\varphi(\varphi(1,0),0)}(n)$$

$$S(n,T(T(0)),1) \approx f_{\varphi(1,0,0)}(n)$$

$$S(n,T(1,0),1) \approx f_{\varphi(1,1,0)}(n)$$

$$S(n,T^{T(0)}(T(1,0)),1) \approx f_{\varphi(1,2,0)}(n)$$

$$S(n,T(1,1),1) \approx f_{\varphi^2(1,1_*,0)}(n)$$

$$S(n,T(1,m + 1),1) \approx f_{\varphi(1,T(1,m),0)}(n)$$

$$S(n,T(1,T(0)),1) \approx f_{\varphi(2,0,0)}(n)$$

$$S(n,T(1,T(m)),1) \approx f_{\varphi(2,0,T(m))}(n)$$

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

$$S(n,T(T(0),T(0)),1) > f_{\varphi(\omega,0,0)}(n)$$

$$S(n,T^{T(0)}(T(0)_*,T(0)),1) > f_{\varphi(1,0,0,0)}(n)$$

$$S(n,T^{S(T(0),1,1)}(T(0)_*,T(0)),1) > f_{\varphi(1,0,0,1)}(n)$$

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

$$S(n,T(1,0_{[m]}),1) > f_{\varphi(1,0_{[m + 1]})}(n)$$

The SeT Function is one of the Fastest Computable functions:

tree(n) function $$\approx f_{\vartheta(\Omega^\omega)}(n)$$

$$S(n,T(1,0_{[T(0)]}),1) > f_{\varphi(1,0_{[\omega]})}(n) = f_{svo}(n) = f_{\vartheta(\Omega^\omega)}(n)$$

$$S(n,T(1,0_{[S(T(0),1,1)]}),1) > f_{\varphi(1,0_{[\omega.2]})}(n)$$

$$S(n,T(1,0_{[S(T(0),1,2)]}),1) > f_{\varphi(1,0_{[\omega.4]})}(n)$$

$$S(n,T(1,0_{[S(T(0),2,1)]}),1) > f_{\varphi(1,0_{[\omega^2]})}(n)$$

$$S(n,T(1,0_{[T(1)]}),1) > f_{\varphi(1,0_{[\varphi(\omega,0)]})}(n)$$

$$S(n,T(1,0_{[T^2(0)]}),1) > f_{\varphi(1,0_{[\varphi(1,0,0)]})}(n)$$

$$S(n,T(1;0),1) = S(n,T(1,0_{[T(1,0)]}),1) > f_{\varphi(1,0_{[\varphi(1,1,0)]})}(n)$$

TREE(n) function $$≥ f_{\vartheta(\Omega^\omega\omega)}(n)$$

$$S(n,T(1;1),1) = S(n,T(1,0_{[T(1;0)]}),1) > f_{\varphi(1,0_{svo})}(n)$$

Large Veblen Ordinal (LVO) $$≥ f_{\vartheta(\Omega^\Omega)}(n)$$

Bird's H(n) function $$\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n)$$

## Further References

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