At 01:15 PM 7/2/2004, you wrote:
At 12:44 02/07/04 -0400, Hal Ruhl wrote:
By the way if some systems are complete and inconsistent will arithmetic be one of them?
As I understand it there are no perfect fundamental theories. So if arithmetic ever becomes complete
then it will be inconsistent.
Yes, if by "arithmetic" you mean an axiomatic system, or a formal theory, or a machine.
No if by arithmetic you mean a set so big that you cannot define it
"define" appears to be a two sided activity. When you define a thing you also define the thing which it is not - a bag of the remainder of "all". Most of the time the latter may not be useful. Since all of arithmetic [and mathematics] is in the Everything and the Everything in my system is the definitional pair to the Nothing, defining the Nothing [or the Everything] automatically defines all of arithmetic along with all of mathematics.
A "Something" is less than the Everything and may or may not contain mathematics or a portion thereof.
in any formal theory,
Well my "theory" seems concerned with the form of its statements that is the "Somethings" and how they alter.
I think my "theory" defines mathematics the way that "The first number that can not be described in less than fourteen words" defines a number that we nevertheless may never actually have in hand.