> On May 21, 10:45 am, "K_h" <KHol...@SX729.com> wrote: >> "Graham Cooper" wrote in message >> >> >> > How many different ways can you order <1,2,3,4....> ? >> >> 2^aleph_0 ways, assuming the axiom of choice and defining the factorial of >> the cardinality of a set S to be equal to the cardinality of the set of all >> bijections of S to itself. >> >> > > The presented algorithm can list all computable permutations of N. > > All 1X2X3X4X5X6X7... of them! > > The set of binary string 2^oo=2X2X2X2X2... > is much smaller.
You can not assume that a property that holds for finite numbers also holds for infinite numbers. For example, n+1 > n for all (finite) naturals, but oo+1 = oo (which I'm sure you agree).
So you can not conclude that n! > 2^n for naturals n>2 implies that c! > 2^c for infinite cardinals c. In fact, for infinite cardinals c, c! = 2^c (if c! is defined as the number of permutations of c and 2^c is defined as the number of subsets of c).