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Definition

$$D[0]$$ is least cardinality and $$D[\alpha+1]$$ is greater cardinality than $$D[\alpha]$$. Also, $$D[\alpha[D_0]]$$ is similar to weakly inaccessible cardinals in OCFs, which:

• $$\psi_{D[\alpha[D_0]]}(0)[1]$$ = $$D[\alpha[1]]$$
• $$\psi_{D[\alpha[D_0]]}(0)[n]$$ = $$D[\alpha[\psi_{D[\alpha[D_0]]}(0)[n-1]]]$$
• $$\psi_{D[\alpha[D_0]]}(\beta+1)[1]$$ = $$D[\alpha[\psi_{D[\alpha[D_0]]}(\beta)+1]]$$
• $$\psi_{D[\alpha[D_0]]}(\beta+1)[n]$$ = $$D[\alpha[\psi_{D[\alpha[D_0]]}(\beta)+\psi_{D[\alpha[D_0]]}(\beta+1)[n-1]]]$$

Then a new definition happen to prove it WAY stronger:

• $$D[\alpha[\beta[D[\alpha[D_{\gamma+1}]]]]]$$ can be reduced to $$D[\alpha[\beta[D_{\gamma}]]]$$, where $$\beta$$ must be less than $$D_{\gamma+1}$$ and $$\gamma$$ > 0.
• Do not follow the rule if $$\beta$$ is a cardinal that does not have the fundamental sequence.

I named the notation $$D[\alpha]$$ as Dropper Ordinal Notation and named $$D_n$$ as (1+n)th dropper cardinal.

Rules

Now here is OCF expansion rules:

• $$\psi_{D[0]}(0)[1]$$ = $$\omega$$
• $$\psi_{D[0]}(0)[n]$$ = $$\omega^{\psi_{D[\alpha]}(0)[n-1]}$$
• $$\psi_{D[0]}(\alpha+1)[1]$$ = $$\psi_{D[0]}(\alpha)$$
• $$\psi_{D[0]}(\alpha+1)[n]$$ = $$\psi_{D[0]}(\alpha)^{\psi_{D[0]}(\alpha+1)[n-1]}$$
• $$\psi_{D[0]}(\alpha)[\beta]$$ = $$\psi_{D[0]}(\alpha[\beta])$$
• $$\psi_{D[\alpha+1]}(0)[1]$$ = $$D[\alpha]$$
• $$\psi_{D[\alpha+1]}(0)[n]$$ = $$D[\alpha]^{\psi_{D[\alpha+1]}(0)[n-1]}$$
• $$\psi_{D[\alpha+1]}(\beta+1)[1]$$ = $$\psi_{D[\alpha+1]}(\beta)$$
• $$\psi_{D[\alpha+1]}(\beta+1)[n]$$ = $$\psi_{D[\alpha+1]}(\beta)^{\psi_{D[\alpha+1]}(\beta+1)[n-1]}$$
• $$\psi_{D[\alpha+1]}(\beta)[\gamma]$$ = $$\psi_{D[\alpha+1]}(\beta[\gamma])$$
• $$\psi_{D[\alpha[D_0]]}(0)[1]$$ = $$D[\alpha[1]]$$
• $$\psi_{D[\alpha[D_0]]}(0)[n]$$ = $$D[\alpha[\psi_{D[\alpha[D_0]]}(0)[n-1]]]$$
• $$\psi_{D[\alpha[D_0]]}(\beta+1)[1]$$ = $$D[\alpha[\psi_{D[\alpha[D_0]]}(\beta)+1]]$$
• $$\psi_{D[\alpha[D_0]]}(\beta+1)[n]$$ = $$D[\alpha[\psi_{D[\alpha[D_0]]}(\beta)+\psi_{D[\alpha[D_0]]}(\beta+1)[n-1]]]$$
• $$\psi_{D[\alpha[D_0]]}(\beta)[\gamma]$$ = $$\psi_{D[\alpha[D_0]]}(\beta[\gamma])$$
• $$\psi_{\alpha}(\beta[\alpha])[1]$$ = $$\psi_{\alpha}(\beta[1])$$
• $$\alpha$$ must be an element of $$D[\beta]$$.
• $$\psi_{\alpha}(\beta[\alpha])[n]$$ = $$\psi_{\alpha}(\beta[\psi_{\alpha}(\beta[\alpha])[n-1]])$$
• $$\alpha$$ must be an element of $$D[\beta]$$.
• $$D[\alpha[\beta[D[\alpha[D_{\gamma+1}]]]]]$$ can be reduced to $$D[\alpha[\beta[D_{\gamma}]]]$$, where $$\beta$$ must be less than $$D_{\gamma+1}$$ and $$\gamma$$ > 0.
• Do not follow the rule if $$\beta$$ is a cardinal that does not have the fundamental sequence.

Fundamental sequences

This is real one to define Dropper Ordinal Notation with fundamental sequences.

• $$\psi_{D[0]}(0)$$ = $$sup\{\omega,\omega^{\omega},\omega^{\omega^{\omega}},...\}$$
• $$\psi_{D[\alpha+1]}(0)$$ = $$sup\{\psi_{D[\alpha]}(0)+1,\omega^{\psi_{D[\alpha]}(0)+1},\omega^{\omega^{\psi_{D[\alpha]}(0)+1}},...\}$$
• $$\psi_{D[A]}(\beta+1)$$ = $$sup\{\psi_{D[A]}(\beta)+1,\omega^{\psi_{D[A]}(\beta)+1},\omega^{\omega^{\psi_{D[A]}(\beta)+1}},...\}$$, only if $$A$$ = 0 or $$\alpha+1$$

More coming soon!

Reduced rules

Now here is the rules using logic:

• $$A_0(\alpha,\beta)$$ = $$\{0\}\cup_{\gamma<\beta}D[\gamma]$$
• $$A_n(\alpha,\beta)$$ = $$\{\gamma+\delta:\gamma,\delta\in A_{n-1}(\alpha,\beta)\}\cup\{\omega^{\gamma}:\gamma\in A_{n-1}(\alpha,\beta)\}$$
• $$A(\alpha,\beta)$$ = $$\cup_{\gamma<\omega}A_{\gamma}(\alpha,\beta)$$

Analysis

You can find an analysis of this OCF vs other OCFs here.