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# B1mb0w

My favorite wikis
• ## The SeT Function

April 9, 2017 by B1mb0w

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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• ## Comparing T() Functions to Ordinals

July 12, 2016 by B1mb0w

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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• ## Omega Count

July 12, 2016 by B1mb0w

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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• ## Growth Rate of the S Function

July 4, 2016 by B1mb0w

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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• ## The Last Alpha Number

June 28, 2016 by B1mb0w

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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