For example, is omega-consistency desirable in a system of formal number theory?
First, I will try to explain what that means. Please ask questions if my explanation is not clear enough for you.
Typically, a system of formal number theory has a symbol 0 for zero, and a one-place operation symbol S where (Sn) represents what we ordinarily write as (n plus one).
Now, suppose that when we plug in a constant value for n the expression P(n) becomes a particular sentence that is true or false for that value of n. Also, suppose that each of the following is a theorem in our system:
(Note that the “etc” is essential, and that it stands for infinitely many theorems).
Then to say that our system is “omega-consistent” is to say that the following isn’t a theorem of our system:
“There exists n such that not P(n).”
That concludes the explanation.
My motivation for starting this thread is a couple of statements from another thread:
“We already have a good set of axioms.”
“Any set of axioms can be used with any grammar rules - provided that there are internally consistent rules.”
Reading over those statements, it is interesting to consider the word “good” in the statement “We already have a good set of axioms.” Suppose that a system of formal number theory that is consistent but not omega-consistent is a good system. It would seem that a system of formal number theory that is omega-consistent is a better system.