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 in any formal theory, like
the set of all true arithmetical sentences. That set cannot be defined in Peano arithmetic for exemple. Some logician use the word "theory" in that generalized sense, but it is misleading.
Now the set of true sentence of arithmetic is that large sense is obviously consistent gieven that it contains only the true proposition! (but you cannot defined it "mechanically").
In the foundation system which I believe contains mathematics from the beginning arithmetic is complete so its inconsistent.
No, because if it is complete, it will not be a mechanical or formal system. Only a theory will
be inconsistent if both complete and enough rich. Not a model.
To borrow Boolos title, I would like to say I get the feeling this list is missing the key road:
Logic, logic and logic ....
BTW an excellent introduction to elementary logic is the penguin book by Wilfried Hodges :
Only the first sentence of the book is false. (will say more on that book later ...)