10,217 Pages

## aka Aarex

My favorite wikis
• I live in USA
• I was born on October 12
• I am Boy
• ## Restarting Sheet Analysis

I am going to restart Sheet Analysis because it's unclear and unorganized as before.

You might check the brand new 'The Sheet Analysis' out here.

• ## Proof of ordinal tetration

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

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

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 $$\omega$$.

Imagine 0 is the empty set and $$\alpha+1$$ is the union set of all elements of $$\alpha$$â€¦

• ## Aarex's Array Notation Rebooted

Here is reboot version of my array notation, simpler than ever.

Growth rate: 0
The simplest and weakest part in AANR (Aarex's Array Notation Rebooted). There is only 1 rule in this extension:

• r(a) = a+1

The limit of simple arrays corresponds to FGH level 0 because r(a) evaluates as the same of $$f_0(a)$$.

Growth rate: $$\omega$$
Let [A]^B = AAA... with Bs, but [A] evaluates, even inside of any [_]s.

• r(a,0) = a+1
• r(a,b) = [r(]aa[,[b-1])]a

If the syntax concentration is too hard for you, here is rules without syntax concentration:

• r(a,0) = a+1
• r(a,b)[0] = a
• r(a,b)[n] = r(r(a,b)[n-1],b-1)[n]
• r(a,b) = r(a,b)[a]

To solve r(2,2) (using without syntax concentration):

• r(2,2) =
• r(2,2)[2] =
• r(r(2,2)[1],1) =
• r(r(r(2,2)[0],1),1) =
• r(r(2,1),1) =
• r(r(2,1)[2],1) =
• r(r(r(2,1)â€¦

• ## DON vs standard ordinal notation

You can find the definition of Dropper Ordinal Notation (DON) here.

Dropper Ordinal Notation Other OCFs
$$D[1]$$ $$\Omega$$
$$D[2]$$ $$\Omega_2$$
$$D[n]$$ $$\Omega_n$$
$$D[\omega]$$ $$\Omega_{\omega}$$
$$D[D[0]]$$ $$\Omega_{\Omega}$$
$$D[D[D[0]]]$$ $$\Omega_{\Omega_{\Omega}}$$
$$\psi_{D[D_0]}(0)$$ $$\psi_I(0)$$
$$\psi_{D[D_0]}(0)^{\psi_{D[D_0]}(0)}$$ $$\psi_I(0)^{\psi_I(0)}$$
$$\psi_{D[\psi_{D[D_0]}(0)+1]}(0)$$ $$\psi_{W\_{\psi_I(0)+1}}(0)$$
$$D[\psi_{D[D_0]}(0)+1]$$ $$\Omega_{\psi_I(0)+1}$$
$$D[\psi_{D[D_0]}(0)+D[\psi_{D[D_0]}(0)+1]]$$ $$\Omega_{\Omega_{\psi_I(0)+1}}$$
$$\psi_{D[D_0]}(1)$$ $$\psi_I(1)$$
$$\psi_{D[D_0]}(D[D_0])$$ $$\psi_I(I)$$
$$D[D_0]$$ $$I$$
$$D[D_0+1]$$ $$\Omega_{I+1}$$
$$D[D_0+D[D_0+1]]$$ $$\Omega_{\Omega_{I+1}}$$
$$D[D_02]$$ $$I_2$$
$$Dâ€¦ Read more > • ## A new, really strong OCF definition? July 29, 2017 by Googleaarex \(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.