Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

70 the mathematical infinityn. If we can do this for a given prime p and every n, we obtain what mathematicianscall a p-adic solution to the problem.It is not difficult to give examples, for any prime number p, of homogeneouspolynomials f of degree d, ind 2 variables, for which the equation f = 0 has nop-adic solutions. On the basis of this and the evidence of several examples, Emil Artinconjectured that if we have more than d 2 variables, then the homogeneous problem ofdegree d always has a p-adic solution.The first big surprise was the proof, by Ax and Kochen in 1965 (Ax and Kochen1965), of Artin’s conjecture with the additional condition that p is larger than somequantity dependent on d but independent of the polynomial f . Their proof relied inan essential way on ZFC and, initially, on the assumption of CH. The second surprisearrived with a construction, by G. Terjanian (Terjanian, G. 1966), of a homogeneouspolynomial of degree 4 in eighteen variables but without 2-adic solutions. Thus, theadditional constraint imposed by Ax and Kochen cannot be avoided. Although a constructiveproof of the Ax-Kochen theorem was eventually found by Paul Cohen (1969),the work by Ax and Kochen showed the power of mathematical logic in tackling, withsuccess, difficult questions in mainstream mathematics.2.10 The Turing Machine and the Busy BeaverThe Turing machine is a famous thought experiment by Alan Turing that shows howlogical calculation works. It is like a typewriter: a tape, infinite in both directions, anda tape head for read and write. It has only two symbols, 1 and 0, representing one bit.The machine, at any given moment, is in one of a possible N states; there is also anadditional special state H , called the halting state. There is a table I of instructions(Sb), as follows: Given a state S and a bit b, the instruction gives a new state Snew, anew bit bnew, and a move M of the tape by one step, which can be only either R to theright or L to the left.The Turing machine works as follows:INITIALIZATION: a tape of symbols (the input) and a table T of instructions (theprogram)RUN:READ the current symbol b on the tapeDO with current state S and instruction SbIF Sb = H then STOPELSEMOVE to state SnewWRITE the symbol bnew on the tapeMOVE the tape one step right or left according to instruction SbCONTINUEEND:The fundamental fact is that such a simple machine can emulate any computer andany finite program. As a very simple example of how such a machine works, considera Turing machine with only one state A and the halting state H , with instruction table