## FANDOM

10,835 Pages

Today, I will define fundamental sequences for my ordinal notation.

## Up to $$\omega$$

These ordinals in this category are the non-negative integers!

If $$\alpha$$ is the integer, then $$\alpha[n] = \alpha$$.

## Up to $$\varepsilon_0$$

Let define $$\omega$$ (the basic ordinal) as:

• $$\omega[n] = n$$

And let define transfinite ordinal as the ordinal which is not equal into any integer.

Then we can define the additive properties of ordinals:

• $$\alpha+\beta$$ = $$\beta$$
• If $$\beta > \alpha$$ and both $$\alpha$$ and $$\beta$$ are any transfinite ordinals.
• Or if $$\alpha$$ is the integer and $$\beta$$ is the transfinite ordinal.
• $$(\alpha+\beta)[n]$$ = $$\alpha+(\beta[n])$$ (If any 2 conditions on the first additive property didn't passed.)

Then we can extend to powers of $$\omega$$:

• $$(\omega^{\alpha}*(\beta+1))[n] = \omega^{\alpha}*\beta+(\omega^{\alpha}[n])$$ ($$\beta$$ must be greater than 1)
• $$\omega^{\alpha+1}[n] = \omega^{\alpha}*n$$ (if $$\alpha + 1$$ is a successor ordinal)
• $$\omega^{\alpha}[n] = \omega^{\alpha[n]}$$ (if $$\alpha$$ is a limit ordinal)

But what's successor ordinal and limit ordinal? A successor ordinal is the ordinal that other ordinal added by 1; and the limit ordinal is the ordinal that NOT a successor ordinal.

## Up to $$\zeta_0$$

• $$\varepsilon_{0}[0] = 1$$
• $$\varepsilon_{\alpha+1}[0] = \varepsilon_{\alpha}+1$$ (if $$\alpha + 1$$ is a successor ordinal)
• $$\varepsilon_{\alpha+1}[n+1] = \omega^{\varepsilon_{\alpha + 1}[n]}$$ (if $$\alpha + 1$$ is a successor ordinal)
• $$\varepsilon_{\alpha}[n] = \varepsilon_{\alpha[n]}$$ (if $$\alpha$$ is a limit ordinal)

Really easy.

## Up to $$\varphi(1,0,0)$$

$$\varphi(0,\alpha)$$ is equal to $$\omega^{1+\alpha}$$; and $$\varphi(1,\alpha)$$ equal to $$\varepsilon_{\alpha}$$

If $$\sigma$$ is any either successor ordinal, limit ordinal, or integer; then $$\sigma+1$$ is the successor ordinal.

• $$\varphi(0,\beta)[0] = \omega^{1+\beta}$$
• $$\varphi(\alpha+1,0)[0] = 0$$
• $$\varphi(\alpha+1,\beta+1)[0] = \varphi(\alpha+1,\beta)+1$$
• $$\varphi(\alpha+1,\beta+1)[n+1] = \varphi(\alpha+1,\varphi(\alpha+1,\beta+1)[n])$$
• $$\varphi(\alpha,0)[n] = \varphi(\alpha[n],0)$$
• $$\varphi(\alpha,\beta+1)[n] = \varphi(\alpha[n],\varphi(\alpha,\beta)+1)$$
• $$\varphi(\alpha,\beta)[n] = \varphi(\alpha,\beta[n])$$

## Up to $$\psi(\Omega^{\Omega^{\omega}})$$

let define $$A$$ as all $$\alpha_{\beta}$$ variables (where $$\beta$$ can be integers and must be less than $$\gamma$$\); and define $$B$$ as all $$\alpha_{\beta}$$ variables (where $$\beta$$ can be integers and must be greater than $$\gamma$$\).

Let $$\alpha_{\delta}$$ will be last $$\alpha_{\beta}$$ variable in any $$\delta+1$$ arguments on $$\varphi()$$. All first arguments of $$\alpha_{\beta}$$ (only contains with 0) can be removed.

• Definition coming soon.

Let define $$\varphi(\alpha+1,n)$$ is $$n+1$$-th term of $$\varphi(\alpha,\_)$$ and $$\varphi(\alpha,n)$$ is $$n+1$$-th term of $$\varphi(\beta,0)$$.

Then $$\varphi(0,A) = \varphi(A)$$ and $$\varphi(...,\alpha+1,0,...n)$$ is $$n+1$$-th term of $$\varphi(...,\alpha,\_,...n)$$

Coming soon.

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