Maybe I should -at least to restate my view of things- add the following remark.
In tetration we do not append exponents to a tower, but bases. So the "partial towers" of an infinite tower are, using a start-value x and a base b
(a) x, b^x, b^b^x, ...^ b^b^x
and not
(b) b, b^b, b^b^...
This is crucial, I think.
For (a) we get then, for instance for base b=sqrt(2) the two solutions
x=2 -> lim h->oo {b,x}^^h =2
x=4 -> lim h->oo {b,x}^^h =4
A supporting argument for this view is also, that if x is already a tower of base b, then the heights are additive...
(c.1) {b,x}^^m={b,{b,y}^^n}^^m = {b,y}^^(m+n)
(c.2) {b,x}^^m={b,{b,{b,{b,y}^^n}^^n}^^n}^^n = {b,y}^^(4*n)
and this is then also coherent with complex fixpoints and real bases as a multisolution problem, even for the limit for infinite heights of towers.
This is also, how the matrix-operator-method works, when used for integer-tetration, although, for the finite integer height we may use associativity to change orders of summation and reflect the approach from the opposite direction.
It is possibly a bit better expressed in my operators-treatise.
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Hmm, to avoid confusion, we should possibly talk of "depth" of a powertower instead of "height" to put the mental focus for the problem at the right side
Gottfried
In tetration we do not append exponents to a tower, but bases. So the "partial towers" of an infinite tower are, using a start-value x and a base b
(a) x, b^x, b^b^x, ...^ b^b^x
and not
(b) b, b^b, b^b^...
This is crucial, I think.
For (a) we get then, for instance for base b=sqrt(2) the two solutions
x=2 -> lim h->oo {b,x}^^h =2
x=4 -> lim h->oo {b,x}^^h =4
A supporting argument for this view is also, that if x is already a tower of base b, then the heights are additive...
(c.1) {b,x}^^m={b,{b,y}^^n}^^m = {b,y}^^(m+n)
(c.2) {b,x}^^m={b,{b,{b,{b,y}^^n}^^n}^^n}^^n = {b,y}^^(4*n)
and this is then also coherent with complex fixpoints and real bases as a multisolution problem, even for the limit for infinite heights of towers.
This is also, how the matrix-operator-method works, when used for integer-tetration, although, for the finite integer height we may use associativity to change orders of summation and reflect the approach from the opposite direction.
It is possibly a bit better expressed in my operators-treatise.
------------------------
Hmm, to avoid confusion, we should possibly talk of "depth" of a powertower instead of "height" to put the mental focus for the problem at the right side
Gottfried
Gottfried Helms, Kassel
