11/13/2007, 08:37 AM
Gottfried Wrote:a) "Tower of height n", n'th tetrate (the latter, if context of continuous operation is focused), heterogeneous, leave "nested" for some more special cases (though I have no idea actually)There are only two kinds of cases with heterogeneous towers / nested exponentials: \( a^{b^c} \) as an expression, and \( x \rightarrow a^{b^x} \) as a function, there are no other special cases. Even if the x is somewhere else, you can view it as the composition of nested exponential functions and other functions. Even the expression \( {e^{ae^{be^c}}} = {(e^a)}^{({e^b)}^{(e^c)}} \) is just a tower (of hight 3 or 4) with weird bases. Reminds me of one paper that tried to call these expressions binary iterated powers but they're still towers to me.
Gottfried Wrote:b) @Jay: top exponent, initial exponent; preferring "initial" to connect to iteration-theory("initial state") and due to the opportunity to be able to better talk about infinite towers and their fixpoints then.I've actually noticed that there is a problem with this. Lets say you have a tower of even hight n, and you want to consider a base/exponent at tower level m=n/2. For all levels > m, level m is a base. For all levels < m, level m is an exponent. So in general what can these be called, if not bases, and not exponents? I like to call them elements or tower elements, but thats just my vote. This comes up so rarely, that it may not make much of a difference. I like the idea of calling them all exponents, but just remember, they are also bases. Also, the levels have been called tower heights, if i remember right.
Andrew Robbins