## FANDOM

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I know how to map ordinals to base-n numbers with my definition. It will split to parts with milestone ordinals only.

## Definition

Expression: ORD → NUM

ORD must be transfinite number and NUM is real numbers only.

## Up to ω

It will be really easy. It only have 1 rule:

• ω → X = X

Where X can be like NUM.

## Up to ε0

This will have some rules because the operators. Let define ● as the rest of the ordinal expression.

• (empty expression) → X = 0
• ● ω → X = ● X → X
• ● ω + n → X = ● ω → X + n
• ● ω x 1 → X = ● ω → X
• ● ω x n → X = ● ω + (ω x n-1) → X
• ● ω ^ 1 → X = ● ω → X
• ● ω ^ n → X = ● ω x (ω ^ n-1) → X
• ε0 → X = ω ^ ω ^ ω ^ ... ^ ω ^ ω ^ ω → X with X ωs

For example: ε0 → 3

• ω ^ (ω ^ ω) → 3
• ω ^ (ω ^ 3) → 3
• ω ^ (ω x ω ^ 2) → 3
• ω ^ (ω x ω x ω ^ 1) → 3
• ω ^ (ω x ω x ω) → 3
• ω ^ (ω x ω x 3) → 3
• ω ^ (ω x (ω + ω x 2)) → 3
• ω ^ (ω x (ω + ω + ω x 1)) → 3
• ω ^ (ω x (ω + ω + ω)) → 3
• ω ^ (ω x (ω + ω + 3)) → 3
• ω ^ (ω + ω x (ω + ω + 2)) → 3
• ω ^ (ω + ω + ω x (ω + ω + 1)) → 3
• ω ^ (ω + ω + ω + ω x (ω + ω)) → 3
• ω ^ (ω + ω + ω + ω x (ω + 3)) → 3
• ...
• ω ^ (ω + ω + ω + ω + ω + ω + ω x ω) → 3
• ω ^ (ω + ω + ω + ω + ω + ω + ω + ω + ω) → 3
• ω ^ (ω + ω + ω + ω + ω + ω + ω + ω + 3) → 3
• ω x ω ^ (ω + ω + ω + ω + ω + ω + ω + ω + 2) → 3
• ω x ω x ω ^ (ω + ω + ω + ω + ω + ω + ω + ω + 1) → 3
• ω x ω x ω x ω ^ (ω + ω + ω + ω + ω + ω + ω + ω) → 3
• ...
• ω x ω x ω x ... x ω x ω x ω → 3 /w 27 ωs
• ω x ω x ω x ... x ω x ω x 3 → 3
• ω x ω x ω x ... x ω x ω + ω x ω x ω x ... x ω x ω x 2 → 3
• ω x ω x ω x ... x ω x ω + ω x ω x ω x ... x ω x ω + ω x ω x ω x ... x ω x ω x 1 → 3
• ω x ω x ω x ... x ω x ω + ω x ω x ω x ... x ω x ω + ω x ω x ω x ... x ω x ω → 3
• ...

## Up to ζ0

Let add the new function, [x], and call it mapping-entry.

Therefore:

• (empty expression) → X = 0
• ● ω → X = ● X → X
• ● ω + n → X = ● ω → X + n
• ● ω x 1 → X = ● ω → X
• ● ω x n → X = ● ω + (ω x n-1) → X
• ● ω ^ 1 → X = ● ω → X
• ● ω ^ n → X = ● ω x (ω ^ n-1) → X
• ● ε0[0] → X = ● 1 → X with X ωs
• ● εe + 1[0] → X = ● εe + 1 → X
• ● εe + 1[n] → X = ● ω ^ εe + 1[n-1] → X
• ● εe + 1 → X = ● εe + 1[X] → X

COMING SOON.