Le 14-mars-06, à 17:28, [EMAIL PROTECTED] (Tom) wrote:

> Another note about numbering. It seems to be that if you repeatedly > make descriptions of descriptions, you eventually end up with all 0's > or all 1's, showing that numbers describing numbers is meaningless. I don't understand. In Peano arithmetic (PA) numbers are usuallu described by the following term build from a language having "0", "s", "(" ")" as primitive symbol. Then the numbers are described by 0, s(0), s(s(0)), s(s(s(0))), s(s(s(s(0)))), etc. For example, PA has axioms: ~(0 = s(x)) and some others. Now you can code "0" by the number 3, s by the number 4, "(" by 5, ")" by 6. And then you can code the finite sequences using the fundamental theorem of arithmetic (which says that prime decomposition of numbers are unique). "s(0)" will be coded by 2^(code of "s") * 3^(code of "(" ) * 5^(code of "0") * 7^(code of ")" ), that is: (2^4)*(3^5)*(5^3)*(7^6) = 6353046000 So 6353046000 is the description of the number one, through that coding, and PA can talk about a *description* of the number one in its own language through the expression: s(s(s(s(s(s(s(s(s(s(s(s ....(0))))))) ....) (with 6353046000 "s" !). Of course, now, a description of a description of "1" will be rather lengthy but I don't see any conceptual problem here(*). By extending that coding it can be shown that PA can handle its own provability abilities, and actually got some impressive self-reference power (indeed described by the modal logic G). (*) In applied computer science more efficient description can be given. In platonist metamathematics, we reason *on* the descriptions without ever really use them, so it is preferable to take conceptually simple one, which are then utterly inefficacious. > Does this also prove that numbers do not have a Platonic existence? Why ? Bruno http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---