GFR Wrote:Could you please explain to me why, from:
a[3]x = x, we should not have, by simple substitutions:
x = a[3](a[3](a[3](a[3]x))), and therefore:
x = a[4]oo ???
Because \( a[4]\infty = \lim_{n\to\infty} \exp_a^{\circ n}(1) \) (note that \( a[4]n = \exp_a^{\circ n}(1) \)) and usually \( a[4]\infty \neq \lim_{n\to\infty} \exp_a^{\circ n}(x) \).
Quote:Concerning your classical example, by putting: a = sqrt(2), we indeed have:
(sqrt(2))^x = x ----> x = (sqrt(2))#oo.
Gianfranco, I exactly explained before that if you put \( x=4 \) this is not true. I really dont know how to explain this even more simple.
The left side is an equation with multiple solutions for \( x \) on the right side of the implication there is a limit it can take only one value (out of the multiple solutions of the left side). In the case of a[4] the right side is the lower real fixed point of \( a^x \), if existing, otherwise it is \( \infty \).
So if you chose the upper real fixed point or a complex fixed point, then the right side is no more true (though the left is).
This really has nothing to do with orthodox or non-orthodox mathematics but with clear thinking.