FANDOM


This blog post sets out the progression of the Wainer ordinal hierarchy used by the Fast-growing hierarchy functions \(f_m(n)\). Technically the equations are not supposed to imply equality of ordinals like \(\omega.3=\omega^2\) . Instead the equations describe how ordinals can be represented differently for particular value of n (n=3 in every case).

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.


\(\omega\) to \(\omega^{\omega}\)

\(\omega\)

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

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

\(\omega^2 + \omega\)

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

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

\(\omega^2.2 + \omega\)

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

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


\(\omega^{\omega}\) to \(\omega^{\omega+1}\) by adding \(\omega^{\omega}\)

\(\omega^{\omega}\)

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

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


\(\omega^{\omega+1}\) to \(\omega^{\omega+2}\) by adding \(\omega^{\omega+1}\)

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

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

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


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

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

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

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


\(\omega^{\omega.2}\) to \(\omega^{\omega.2+1}\) by adding \(\omega^{\omega.2}\)

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

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

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


\(\omega^{\omega.2+1}\) to \(\omega^{\omega.2+2}\) by adding \(\omega^{\omega.2+1}\)

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

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

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


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

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

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

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


\(\omega^{\omega^2}\) to \(\omega^{\omega^2+1}\) by adding \(\omega^{\omega^2}\)

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

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

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


\(\omega^{\omega^2+1}\) to \(\omega^{\omega^2+2}\) by adding \(\omega^{\omega^2+1}\)

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

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

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


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

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

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

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


\(\omega^{\omega^2+\omega}\) to \(\omega^{\omega^2+\omega+1}\) by adding \(\omega^{\omega^2+\omega}\)

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

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

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


\(\omega^{\omega^2+\omega+1}\) to \(\omega^{\omega^2+\omega+2}\) by adding \(\omega^{\omega^2+\omega+1}\)

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

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

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


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

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

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

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


\(\omega^{\omega^2+\omega.2}\) to \(\omega^{\omega^2+\omega.2+1}\) by adding \(\omega^{\omega^2+\omega.2}\)

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

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

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


\(\omega^{\omega^2+\omega.2+1}\) to \(\omega^{\omega^2+\omega.2+2}\) by adding \(\omega^{\omega^2+\omega.2+1}\)

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

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

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

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

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

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

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


\(\omega^{\omega^2.2}\) to \(\omega^{\omega^2.2+1}\) by adding \(\omega^{\omega^2.2}\)

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

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

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


\(\omega^{\omega^2.2+1}\) to \(\omega^{\omega^2.2+2}\) by adding \(\omega^{\omega^2.2+1}\)

\(\omega^{\omega^2.2+1}\)

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

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


\(\omega^{\omega^2.2+2}\) to \(\omega^{\omega^2.2+\omega}\) by adding \(\omega^{\omega^2.2+2}\)

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

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

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


\(\omega^{\omega^2.2+\omega}\) to \(\omega^{\omega^2.2+\omega+1}\) by adding \(\omega^{\omega^2.2+\omega}\)

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

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

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


\(\omega^{\omega^2.2+\omega+1}\) to \(\omega^{\omega^2.2+\omega+2}\) by adding \(\omega^{\omega^2.2+\omega+1}\)

\(\omega^{\omega^2.2+\omega+1}\)

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

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


\(\omega^{\omega^2.2+\omega+2}\) to \(\omega^{\omega^2.2+\omega.2}\) by adding \(\omega^{\omega^2.2+\omega+2}\)

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

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

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


\(\omega^{\omega^2.2+\omega.2}\) to \(\omega^{\omega^2.2+\omega.2+1}\) by adding \(\omega^{\omega^2.2+\omega.2}\)

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

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

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


\(\omega^{\omega^2.2+\omega.2+1}\) to \(\omega^{\omega^2.2+\omega.2+2}\) by adding \(\omega^{\omega^2.2+\omega.2+1}\)

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

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

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


\(\omega^{\omega^2.2+\omega.2+2}\) to \(\epsilon_0\) by adding \(\omega^{\omega^2.2+\omega.2+2}\)

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

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

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

\(= \omega^{\omega^2.2+\omega.\omega} = \omega^{\omega^2.2+\omega^2} = \omega^{\omega^2.3} = \omega^{\omega^{\omega}} = \epsilon_0\)

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