10,256 Pages

## No Longer RelevantEdit

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.

## Omega CountEdit

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 various values of $$n$$:

ordinal $$n=2$$ $$n=3$$ $$n=4$$
$$\omega$$ $$\Omega(2)$$ $$\Omega(2)$$ $$\Omega(2)$$
$$\epsilon_0$$ $$\Omega(2^2)$$ $$\Omega(3^{3^3})$$ $$\Omega(4^{4^{4^4}})$$
$$\epsilon_1$$ $$\Omega(4^4)$$ $$\Omega(3^{3^3}\uparrow\uparrow 2)$$ $$\Omega(4^{4^{4^4}}\uparrow\uparrow 2)$$

Here are the Omega Counts for more ordinals which uses the following:

• $$n=2$$
• $$x$$ to represent the number from the previous cell in the table
• $$u()$$ to represent $$u(x) = x\uparrow\uparrow 2$$:
$$\varphi(row,column)$$ $$0$$ $$1$$ $$\omega$$ $$\omega + 1$$
$$1$$ $$\Omega(4)$$ $$\Omega(4^4)$$ $$\Omega(u^2(4))$$ $$\Omega(u^3(4))$$
$$\omega$$ $$\Omega(u^4(4))$$ $$\Omega(u^6(4))$$ $$\Omega(u^8(4))$$ $$\Omega(u^{10}(4))$$
$$\omega + 1$$ $$\Omega(u^{12}(4))$$ $$\Omega(u^{16}(4))$$ $$\Omega(u^{20}(4))$$ $$\Omega(u^{24}(4))$$