## FANDOM

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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 $$\omega^{\omega^2.2+\omega.2+2}$$

$$\omega^{\omega^2.2+\omega.2+2}$$

$$\omega^{\omega^2.2+\omega.2+2} + \omega^{\omega^2.2+\omega.2+2} = \omega^{\omega^2.2+\omega.2+2}.2$$

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

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