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# B1mb0w

My favorite wikis
• ## 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 varioâ€¦

• ## 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 refâ€¦

• ## 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 inâ€¦

• ## Alpha 1 Ruler Function

June 28, 2016 by B1mb0w

The Alpha Function can generate every finite integer up to a very large number. The Alpha Function has a growth rate faster than $$f_{LVO}(n)$$ for any n.

The Alpha Function is 'calibrated' to accept real number inputs up to 10,000 and generate unique S() function outputs representing any and every big number up to the size of $$f_{LVO}(v)$$ for any n.

I have started work on a set of Ruler Functions based on the Alpha function that will be useful to measure the size of very large numbers. Different 'rulers' can be used as required. A rough sketch of how this will look is:

$$\alpha_0$$ Ruler Function

$$\alpha_0(100) = \alpha(31.6) >> f_{svo}(3)$$ approximately

$$\alpha_1$$ Ruler Function

$$\alpha_1(100) = \alpha(5.79955) = f_{\omega + 1}(64) >> g_{64} =â€¦ Read more > • ## Alpha 0 Ruler Function June 27, 2016 by B1mb0w The Alpha Function can generate every finite integer up to a very large number. The Alpha Function has a growth rate faster than \(f_{LVO}(n)$$ for any n.

The Alpha Function is 'calibrated' to accept real number inputs up to 10,000 and generate unique S() function outputs representing any and every big number up to the size of $$f_{LVO}(v)$$ for any n.

I have started work on a set of Ruler Functions based on the Alpha function that will be useful to measure the size of very large numbers. Different 'rulers' can be used as required. A rough sketch of how this will look is:

$$\alpha_0$$ Ruler Function

$$\alpha_0(100) = \alpha(31.6) >> f_{svo}(3)$$ approximately

$$\alpha_1$$ Ruler Function

\(\alpha_1(100) = \alpha(5.79955) = f_{\omega + 1}(64) >> g_{64} =â€¦