FANDOM


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.

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