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Ciao,

I just like recall an 'arithmatical expression' discussed by Euler in onenof his books (I'll be more precise a second time, possibily). The expression is (x^x). I hope BillFish will not be hurt id i link x^x to is b^x, tinking of the first as a genralization of the second.

Euler spotted many peculiar propreties of x^x.

1) also if is somehow "plottable" as y=x^x, the curves is almoust everywere indetermined in its codomain, (y).

Except for Natural Numbers, already the Rationals gives undecidable results, in particular if they ar unlimited fractions. In this case, the question about the nature of the output of x^x, will requires singular case study for each x, or finding a general theorem in Numer Theory that states once for all to which class this ouput X^X belongs.

Whit Irrationals, the case is even more obvoius, since if had already by a great deal of work demonstrating the irrationality of 2^{1/2}, 5^{1/2}, 7^{1/2}, 2^{1/3}, 11^{1/4 }and of the sum 2^{1/2}+3^{1/2} .... and not much further than this (cfr. Hardy & Wright, Oxford 1978), it is very plausible how problematic will be determining the nature of (x irr) ^ ^{(x irr)}. Which means that for the Irrationals the curve is maynly indetermined.

Finally taking into account the Trascendental numbers the question gets even deeper for the reason that, though they have the same Cardinal of the continuum, we know really a little about them. The numbers "e", "Pi", "ln2", "ln3/ln2", "e^{ Pi }", "2^{^(1/2) }", J_{0}(1) had been shown tey areTrascendental, but not jet "2^{^ e} ", "2^{^ Pi} ", "Pi^{^ e "}, or the Euler's constant "gamma". For the Trascendental we can say the the curve is twice indetermined, knowing just a few of the numbers and so of the x on which something can be said as comes uot to determine the nature of y=x^x.

Euler goes into alla possible combinations amongst the clases of numbers. To show as x^x si such a peculiar mathematical object.

I think for any of you which master Maple tis "object" could be an interesting to get a plot (not by approximation), and I bet the proper realm of parametrization will be the complex one. The reason I say this comes from the mapping propreties of x^x, especially if one takes the combinatorics of N, Q, Z, T, lets say x natural over x rational the simplest expression that already entrains irrational as x^x, or even trascendental, for certain numerical combination that possibily could'nt fit in the original imput classes.

The Curve y=x^x over R, could be also intriguing for the fact tha it seams nowere dense, (always not taking into consideration the approximation wich allow us to see it as a continuos line, iperexponential also). Esplicitly the mesure of the line, over a limited sets of values, must be fractal.

Taht's it.

This I think can explain, peraphs, the behawiour of "a+b^x" that surprize BillFish, and the aidea that this alla might have to do with God. On a different step, Goedel profess atheist, gave a go him too to the Onthological Pouf of Gog, using modal Logic, and a similar approach he used to answer definitively at Rieman hypotesis on artmetization of Mathematics (as you know the answer was "no"). Cantor too, by setting is Meditation on Transfinite Numbers, felt his work very closed to a quest for God.

So BillFish, you are in a very good company!!!

Take care

Federiclet

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