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in: Buchholz's psi-functions

3-ary Psi calculator

Koteitan
Koteitan • 9 January 2021 • User blog:Koteitan

Japanese version of the post

The site[]

I made an online calculator for the expansion of 3-ary Psi.

https://koteitan.github.io/ordex/

Expandable notations[]

The notations which can be expanded with the site are the followings:

Notation[]

The notation can be expressed by the five charactors "0", "(", ")", ",", "+" and the elements of the following \(T\) can be recognized. \begin{eqnarray} 0 &\in&T\\ (X_1,X_2,X_3)&\in&T\leftarrow X_1 \in T \land X_2 \in T \land X_3 \in T \\ (2 \leq \forall m \lt \omega)(X_1+\cdots+X_m&\in&T\leftarrow X_1 \in T \land X_2 \in T \land X_3 \in T \leftarrow (1\leq \forall n\leq m X_1\in PT))\\ (X_1,X_2,X_3)&\in&T\leftarrow X_1 \in PT \land X_2 \in T \land X_3 \in T \end{eqnarray} For example, 

0
(0,0,0)
(0,0,0)+(0,0,0)
(0,0,0)+(0,0,0)+(0,0,0)
(0,0,(0,0,0))
(0,0,(0,(0,0,0),0))
(0,0,(0,(0,0,(0,0,0))+(0,0,0),0))
((0,0,0),0,0)

.

"parse X" button let you confirm how the system recognized your input. "tree X" button let you confirm how the tree structure of your input.

You can use the following syntax sugar as well:

  • 1 = (0,0,0) = \(1\)
  • 2 = (0,0,0)+(0,0,0) = \(2\)
  • 3 = (0,0,0)+(0,0,0)+(0,0,0) = \(3\)
  • ... similar up to 9
  • w = (0,0,(0,0,0)) = \(\omega\)
  • W = (0,(0,0,0),0) = \(\Omega\)
  • e = (0,0,W) = \(\varepsilon_0\)
  • z = (0,0,(0,1,W)) = \(\zeta_0\)
  • G = (0,0,(0,1,(0,1,W))) = \(\Gamma_0\)


Fundamental sequence expansion feature[]

The main feature of the calculator is the expansion of the fundamental sequence. Write notations into into the left side and press "expand X[n]" button. The calculator expand it and output the result to right side. 

X Y

means X[Y].

Examples: 

w 3
w+5 3
(0,0,(0,0,w+5)) 3
(0,0,(0,0,w+w+2)) 3
(0,0,W) 3
(0,0,(0,1,W)) 3
(0,0,(0,W,0)) 3
(0,0,(W,0,0)) 3

↓ 

3
w+1+1+1+1
(0,0,(0,0,w+1+1+1+1)+(0,0,w+1+1+1+1)+(0,0,w+1+1+1+1))
(0,0,(0,0,w+w+1)+(0,0,w+w+1)+(0,0,w+w+1))
(0,0,(0,0,(0,0,w)))
(0,0,(0,1,(0,0,(0,1,(0,0,(0,1,(0,0,(0,1,1))))))))
(0,0,(0,(0,0,(0,(0,0,(0,e,0)),0)),0))
(0,0,((0,0,((0,0,((0,0,(1,0,0)),0,0)),0,0)),0,0))


Other feature[]

  • compare: Write as X Y and press "compare X<Y" button. The calculator will compare them and output the result.
  • dom: Write as X and press "dom(X)" button. The calculator output the value of dom(X).
  • You can add the multiple notations with the space charactors in the right of the one as well. You can add the another notations into the next line as well. It will calculate them at once. The pairs are calculated with the expand and comparison. Please use to get the multiple results at onces.


Example of analysis[]

The followings are the examples I confirmed the result of expansion.

The followings are the examples for the expansion by [3] of the all notations on the page.

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