User blog:B1mb0w/From omega to epsilon nought

This blog post sets out the progression of the Wainer ordinal hierarchy used by the Fast-growing hierarchy functions \(f_m(n)\) and assumes n = 3 for all the worked examples.

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

\(\omega^2\) to \(\omega^2+\omega.2\)
\(\omega^2\)

\(\omega^2+1\)

\(\omega^2+2\)

\(\omega^2+3 = \omega^2+\omega\)

\(\omega^2+\omega+1\)

\(\omega^2+\omega+2\)

\(\omega^2+\omega+3 = \omega^2+\omega+\omega = \omega^2+\omega.2\)

\(\omega^2+\omega.2\) to \(\omega^{\omega} + \omega^2\) by adding \(\omega\)
\(\omega^2+\omega.2\)

\(\omega^2+\omega.2 + \omega = \omega^2+\omega.3 = \omega^2+\omega.\omega = \omega^2+\omega^2 = \omega^2.2\)

\(\omega^2.2+\omega\)

\(\omega^2.2+\omega.2\)

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

\(\omega^{\omega} + \omega\)

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

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

\(\omega^{\omega} + \omega^2\) to \(\omega^{\omega}.\omega\) by adding \(\omega^2\)
\(\omega^{\omega}+\omega^2\)

\(\omega^{\omega}+\omega^2.2\)

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

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

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

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

\(\omega^{\omega}.\omega\) to \(\omega^{\omega}.\omega^2\) by adding \(\omega^{\omega}\)
\(\omega^{\omega}.\omega\)

\(\omega^{\omega}.\omega+\omega^{\omega} = \omega^{\omega}.(\omega+1)\)

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

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

\(\omega^{\omega}.\omega.2+\omega^{\omega} = \omega^{\omega}.(\omega.2+1)\)

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

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

\(\omega^{\omega}.\omega^2\) to \(\omega^{\omega^2}\) by adding \(\omega^{\omega}.\omega^2\)
\(\omega^{\omega}.\omega^2\)

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

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

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

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

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

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

From \(\omega^{\omega^2}\) to \(\omega^{\omega^2} + \omega^{\omega.2}.\omega.2\) by adding \(\omega^{\omega.2}\)
\(\omega^{\omega^2}\)

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

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

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

\(\omega^{\omega^2} + \omega^{\omega.2}.\omega + \omega^{\omega.2} = \omega^{\omega^2} + \omega^{\omega.2}.(\omega+1)\)

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

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

From \(\omega^{\omega^2} + \omega^{\omega.2}.\omega.2\) to \(\epsilon_0\) by adding \(\omega^{\omega.2}.\omega\)
\(\omega^{\omega^2} + \omega^{\omega.2}.\omega.2\)

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

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

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

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

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

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

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

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

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