Le 23-mai-06, à 06:57, George Levy a écrit :

One can create faster and faster rising functions and larger and larger number until one is blue in the face. The point is that no matter how large a finite number n one defines, I can stand on the shoulder of giants and do better by citing n+1 using simple addition.

Now if somehow one came up with a finite number n so large that I am not allowed to say n+1 as if I was up against an overflow limitation similar to that found in computers, then there would be no physical way for me to invent or cite a larger number.  So it seems that if we are to define a largest finite number we must define it in conjunction with the number b of bits that we are allowed to use to express this number. For a given number of bits b the largest number would be n(b).

If we use the Ackerman series of functions we need 1 bit for addition, 2 bits for multiplication, 3 bits for exponentiation, 4 bits for tetration etc... These bits are required in addition to the bits for the input parameter(s) of the function.

What is the largest number of bits which are available to me to define an Ackerman function or some other fast rising function? Possibly the number of particles in the universe? I  don't know if the fairy would be satisfied or if I could personally herd all those bits.

The fairy gives you some amount of papers. She is fair enough to provide more if you ask politely ;)
The goal is to name a finite but as huge as possible number. It does not (obviously) consist in naming the biggest number (which does not exist as your little reasoning above shows clearly), nor does it consist in writing the best possible solution with respect to an available number of bits, (although we *will* arrive at this (much less simple) problem later).

Is she expecting me to hand in a piece of paper with the number written on it? Maybe then the answer would be the number generated by the largest Ackerman function that I can write with a very fine pen on this piece of paper.

Actually this can be considered as a good answer in the sense that the Ackermann number are already unimaginably gigantic, but that's nothing compared to the number which we will obtain by diagonalizations. Remember that my goal is to explain "diagonalization". Actually, the goal of this thread is to explain Smullyan's "heart of the matter" in his FU book. For this we need not only to understand diagonalisation, but we will need to understand varieties of effective (programmable) and non effective diagonalizations before. I'm a bit sorry for the work I'm asking you ...



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