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In another thread I found interesting expressions (y is a result of those operations, whatever number type it may take):
y=0[0]0
y=1[1]1 = 1
y=2[2]2 = 2*2=4
y=3[3]3 = 3^3^3 = 19683
y=4[4]4=?
.....
y=n[n]n
......
y=x[x]x
......
y= i[ i]i
......
y=z[z]z
(this may be left out : y= q[q]q where q is quaternion...)
The question is, numbers 1, 2, 3, 4, ..n, ...x, i. , z .... q... are "something" - inverted hyperoperations of y.
How should we call this something and can we find these given y?
y=x[x]x, what is x if y is given and how many x will fit the equation?
Ivars
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If we limit ourselves to y = n[n]n, with n natural, the sequence is known and it is called: "The Ackermann Sequence". See:
http://www.scottaaronson.com/writings/bignumbers.html
I suppose it is so, if I am not confusing things. The numbers so obtained are the "Ackermann numbers". Very big .... !
GFR
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04/27/2008, 02:59 PM
(This post was last modified: 04/27/2008, 04:18 PM by GFR.)
Dear Ivars:
0[0]0 = ??? (I try to avoid troubles ...  )
1[1]1 = 1+1 = 2
2[2]2 = 2*2 = 4
3[3]3 = 3^3 = 27
4[4]4 = 4#4 = 4^[4^[4^4]]] .... uuuuuhhhhh !!!
......
oo[w]oo = ...... infinite-omega-infinite (Qickfur said that it is still enumerable! I think he is right!). Actually, we could write it as oo[oo]oo, but I think that the entity between the brackets is an ordinal. Am I wrong? Maybe yes !  Ops, suicide comment: how about s=0, the 0-th hyperop?
GFR
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Therefore, the Ackermann numbers can be noted as: y = Ack(n) = n[n]n, where "n" is the order of Ack(n). If Ack(n) is a function of n, then n = InvAck(y) is its inverse.
GFR
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04/27/2008, 04:42 PM
(This post was last modified: 04/27/2008, 05:11 PM by Ivars.)
Hi GFR
Thanks . And thanks for corrections as well. Sometimes I am too lazy just too learn the basics to the last detail...
But finally I understood what is Ackermann function as well, in Your clear and simple bracket notation. Quite impressive function.Very fast. And its Inverse seems to be very slow.
But now suddenly I am interested in i[i]i = Ack[i]. What shall I do 
Ivars
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Mmmmm!! Let's think about that. And how about Ack(0). Does it give 0, 1, 2 (?!) or indeterminate or NAN (Not A Number). Bah, time will solve all that
GFR
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04/28/2008, 09:13 AM
(This post was last modified: 04/28/2008, 09:15 AM by Ivars.)
And Ack (-1) , Ack (-n) Ack (-x) , Ack (-I) ..... as well.
What was Ack (+-oo) = +-oo[+-oo]+-oo? The same poor single infinity oo which has to hold so much into it? Can not believe it.
Ivars
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05/02/2008, 06:10 PM
(This post was last modified: 05/02/2008, 06:12 PM by andydude.)
No, the Ackermann numbers ( from MathWorld) are defined as \( n[n+2]n \), because they are defined by arrow notation, and not box notation.
The numbers defined by \( n[n]n \) have no name.
Andrew Robbins
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05/02/2008, 10:33 PM
(This post was last modified: 05/02/2008, 10:36 PM by GFR.)
Dear Andrew!
concerning:
andydude Wrote:No, the Ackermann numbers (from MathWorld) are defined as \( n[n+2]n \), because they are defined by arrow notation, and not box notation.
The numbers defined by \( n[n]n \) have no name. Actually, in the Department of Engineering and Computer Science of the MIT there is a "research line" covering a rapidly increasing sequence of numbers, called " The Ackermann Sequence", defined as :
Ack(n) = n[n]n.
Of course, the much better known sequence defined by the Knuth's up-arrows notation is starting by the exponentiation rank 1[3]1 = 1^1 = 1 and it defines the " Sequence of the Ackermann Numbers", something like:
An(n) = n[n+2]n.
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. The second and better known version is strongly influenced by the Knuth's arrow notation and gives a kind of ultra-exponential sequence, completely ignoring hyperranks 1 and 2 (not to mention ... 0!).
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)).
As we can see the situation is both extremely clear and very confused, from the terminological point of view.
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.
GFR
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GFR Wrote: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.
Quote: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.
Quote: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!
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