Le 14-mars-06, à 17:28, daddycaylor.domain.name.hidden (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/
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Received on Wed Mar 15 2006 - 06:24:57 PST