User blog:B1mb0w/From omega to epsilon nought

From \(f_{\omega^2}(3)\) to \(f_{\epsilon_0}(3)\)

This progression is used to build bigger numbers in the Strong D Function blog post. If anybody can find errors, I'd appreciate your help to understand the correct progression.

Progression \(\omega\): From \(f_{\omega^2}(3)\) to \(f_{\omega^2+\omega}(3)\)
\(f_{\omega^2}(3)\)

Step 1 \(f_{\omega^2}^2(3)\)

Step 2 \(f_{\omega^2}^3(3) = f_{\omega^2+1}(3)\)

Step 3 \(f_{\omega^2+1}^2(3)\)

Step 4 \(f_{\omega^2+1}^3(3) = f_{\omega^2+2}(3)\)

Step 5 \(f_{\omega^2+2}^2(3)\)

Step 6 \(f_{\omega^2+2}^3(3) = f_{\omega^2+3}(3) = f_{\omega^2+\omega}(3)\)

From \(f_{\omega^2+\omega}(3)\) to \(f_{\omega^2.2}(3)\)
Each step here assumes all of progression \(\omega\) is repeated:

\(f_{\omega^2+\omega}(3)\)

\(f_{\omega^2+\omega+\omega}(3) = \(f_{\omega^2+\omega.2}(3)\)

\(f_{\omega^2+\omega.2+\omega}(3) = f_{\omega^2+\omega.3}(3) = f_{\omega^2+\omega^2}(3) = f_{\omega^2.2}(3)\)

Progression \(\omega^2\): From \(f_{\omega^2.2}(3)\) to \(f_{\omega^{\omega}}(3)\)
Each step here assumes all of progression \(\omega\) is repeated:

\(f_{\omega^2.2}(3)\)

Step 1 \(f_{\omega^2.2+\omega}(3)\)

Step 2 \(f_{\omega^2.2+\omega+\omega}(3) = f_{\omega^2.2+\omega.2}(3)\)

Step 3 \(f_{\omega^2.2+\omega.2+\omega}(3) = f_{\omega^2.2+\omega.3}(3) = f_{\omega^2.2+\omega^2}(3) = f_{\omega^2.3}(3) = f_{\omega^3}(3) = f_{\omega^{\omega}}(3)\)

Progression \(\omega^{\omega}\): From \(f_{\omega^{\omega}}(3)\) to \(f_{\omega^{\omega}.2}(3)\)
Each step here assumes all of progression \(\omega^2\) is repeated:

\(f_{\omega^{\omega}}(3)\)

Step 1 \(f_{\omega^{\omega}+\omega^2}(3)\)

Step 2 \(f_{\omega^{\omega}+\omega^2.2}(3)\)

Step 3 \(f_{\omega^{\omega}+\omega^2.3}(3) = f_{\omega^{\omega}+\omega^3}(3)= f_{\omega^{\omega}+\omega^{\omega}}(3) = f_{\omega^{\omega}.2}(3)\)

From \(f_{\omega^{\omega}.2}(3)\) to \(f_{\epsilon_0}(3)\)
Each step here assumes progression \(\omega^{\omega}\) is repeated:

\(f_{\omega^{\omega}.2}(3)\)

\(f_{\omega^{\omega}.3}(3) = f_{\omega^{\omega}.\omega}(3)\)

\(f_{\omega^{\omega}.(\omega+1)}(3)\)

\(f_{\omega^{\omega}.(\omega+2)}(3)\)

\(f_{\omega^{\omega}.(\omega+3)}(3) = f_{\omega^{\omega}.(\omega.2)}(3)\)

\(f_{\omega^{\omega}.(\omega.2+1)}(3)\)

\(f_{\omega^{\omega}.(\omega.2+2)}(3)\)

\(f_{\omega^{\omega}.(\omega.2+3)}(3) = f_{\omega^{\omega}.(\omega^2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+1)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+3)}(3) = f_{\omega^{\omega}.(\omega^2+\omega)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+\omega+1)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+\omega+2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+\omega+3)}(3) = f_{\omega^{\omega}.(\omega^2+\omega.2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+\omega.2+1)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+\omega.2+2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2+\omega.2+3)}(3) = f_{\omega^{\omega}.(\omega^2+\omega.2+\omega)}(3) = f_{\omega^{\omega}.(\omega^2+\omega.3)}(3) = f_{\omega^{\omega}.(\omega^2+\omega^2)}(3) = f_{\omega^{\omega}.(\omega^2.2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+1)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+3)}(3) = f_{\omega^{\omega}.(\omega^2.2+\omega)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+\omega+1)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+\omega+2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+\omega+3)}(3) = f_{\omega^{\omega}.(\omega^2.2+\omega.2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+\omega.2+1)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+\omega.2+2)}(3)\)

\(f_{\omega^{\omega}.(\omega^2.2+\omega.2+3)}(3) = f_{\omega^{\omega}.(\omega^2.2+\omega.2+\omega)}(3) = f_{\omega^{\omega}.(\omega^2.2+\omega.3)}(3) = f_{\omega^{\omega}.(\omega^2.2+\omega^2)}(3) = f_{\omega^{\omega}.(\omega^2.3)}(3) = f_{\omega^{\omega}.(\omega^3)}(3) = f_{\omega^{\omega^2}}(3)\)

...Work In Progress...

\(f_{\omega^{\omega}.\omega^2.3}(3) = f_{\omega^{\omega}.\omega^3}(3) = f_{\omega^{\omega}.\omega^{\omega}}(3) = f_{\omega^{\omega^2}}(3)\)

\(f_{\omega^{\omega^2}.\omega^{\omega}}(3)\)

\(f_{\omega^{\omega^2}.\omega^{\omega}.2}(3)\)

\(f_{\omega^{\omega^2}.\omega^{\omega}.3}(3) = f_{\omega^{\omega^2}.\omega^{\omega}.\omega}(3)\)

\(f_{\omega^{\omega^2}.\omega^{\omega}.\omega.2}(3)\)

\(f_{\omega^{\omega^2}.\omega^{\omega}.\omega.3}(3) = f_{\omega^{\omega^2}.\omega^{\omega}.\omega^2}(3)\)

\(f_{\omega^{\omega^2}.\omega^{\omega}.\omega^2.2}(3)\)

\(f_{\omega^{\omega^2}.\omega^{\omega}.\omega^2.3}(3) = f_{\omega^{\omega^2}.\omega^{\omega}.\omega^3}(3) = f_{\omega^{\omega^2}.\omega^{\omega}.\omega^{\omega}}(3) = f_{\omega^{\omega^2}.\omega^{\omega}.\omega^{\omega}}(3)\)