## FANDOM

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Here is the proof that $$^{\omega 2}\omega$$ = $$\varepsilon_{\varepsilon_0}$$ or a weaker ordinal $$\varepsilon_1$$.

## Ordinal tetration

Ordinal tetration is a type of normal form but we can express ordinals with 0, 1, $$\omega$$, addition, multiplication, exponentiation, and tetration.

## Proof

The proof require to have 1 new property, OPTD ($$\omega$$ power tower depth). Now what's OPTD? A natural number have OPTD 0, $$\omega$$ to maximum value that less than $$\omega^{\omega}$$ have OPTD 1, and $$\omega^{\alpha}$$ have OPTD $$\beta+1$$, such that $$\alpha$$ have OPTD $$\beta$$.

Also, I also show that $$1+\omega$$ = $$\omega$$ and $$\omega^{\varepsilon_02}$$ have OPTD $$\omega$$.

### Sub-proof 1 ($$1+\omega$$ = $$\omega$$)

Imagine 0 is the empty set and $$\alpha+1$$ is the union set of all elements of $$\alpha$$ and the set of $$\alpha$$. Therefore $$\omega$$ = $$\{,\{\},\{,\{\}\},\{,\{\},\{,\{\}\}\},...\}$$.

However, we start at 1 and increase by 1 each till we reach $$1+\omega$$. The result is unexpected and results as the set of $$\omega$$. Therefore, I proved that $$1+\omega$$ = $$\omega$$.