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 \(u(x) = x\uparrow\uparrow 2\):
\(\varphi(row,column)\) \(0\) \(1\) \(\omega\) \(\omega + 1\)
\(1\) \(\Omega(4)\) \(\Omega(4^4)\) \(\Omega(u^2(4))\) \(\Omega(u^3(4))\)
\(\omega\) \(\Omega(u^4(4))\) \(\Omega(u^6(4))\) \(\Omega(u^8(4))\) \(\Omega(u^{10}(4))\)
\(\omega + 1\) \(\Omega(u^{12}(4))\) \(\Omega(u^{16}(4))\) \(\Omega(u^{20}(4))\) \(\Omega(u^{24}(4))\)