Name for sequence y=0[0]0. y=1[1]1, y=2[2]2, y= n[n]n, y=x[x]x, y=i[i]i, y=z[z]z etc?

[bo198214]

In my opinion, the first sequence (the Ackermann Sequence, proposed by Prof. Scott Aaronson, MIT) is more compatible with the subject that we are studying.

Indeed I wondered whether there are different definitions of Ackermann numbers out there.

It always goes without saying that the "Ackermann Function" is a function of two variables (attention please: not a two-valued function, because they are strictly forbidden in this Forum), noted as:
A(0, n) = n+1
A(m, 0) = A(m-1, 1)
A(m, n) = A(m-1, A(m, n-1)).

But be careful, this is what novadays is called Ackermann function. Its main purpose is a simplification of the original Ackermann function - which was indeed a 3 argument function and starting with addition as 0th operation - for proving that there are recursive functions that are not primitive recursive. You can read about the original Ackermann function, i.e. the function that Ackermann defined himself, in his article [1].

[1] Wilhelm Ackermann (192. "Zum Hilbertschen Aufbau der reellen Zahlen". Mathematische Annalen 99: 118–133.

Unfortunately, due to other personal and family priorities, from to-day I am obliged to give much lesser time to these important and intersting subjects.

It was nice to discuss with you.

Hope we see you again here soon!