Perhaps LessWrong is a place where I can say “Your question is wrong” without causing unintended offense. (And none is intended.)
Yes, for ω-consistency to even be defined for a theory it must interpret the language of arithmetic. This is a necessary precondition for the statement you quoted, and does not contradict it.
Work in PA, and take a family of statements P(n) where each P(n) is true but independent of PA, and not overly simple statements themselves—say, P(n) is “the function epsilon n in the fast growing hierarchy is a total function”. (The important thing here is that the statement is at least Pi2---true pure existence statements are always provable, and if the statements were universal there would be a different ω-consistency problem. The exact statement isn’t so important, but not that these statements are true, but not provable in PA.)
Now consider the statement T=”there is an n such that P(n) is false”. PA+T has no standard model (because T is false), but PA+T doesn’t prove any of the P(n), let alone all of them, so there’s no ω-consistency problem.
I’m a fan of Enderton’s “A Mathematical Introduction to Logic”. It’s short and very precisely written, which can make it a little difficult to learn from on its own, but together with a general familiarity with the subject and using wikipedia for additional examples elaboration, it should be perfect.
Perhaps LessWrong is a place where I can say “Your question is wrong” without causing unintended offense. (And none is intended.)
Yes, for ω-consistency to even be defined for a theory it must interpret the language of arithmetic. This is a necessary precondition for the statement you quoted, and does not contradict it.
Work in PA, and take a family of statements P(n) where each P(n) is true but independent of PA, and not overly simple statements themselves—say, P(n) is “the function epsilon n in the fast growing hierarchy is a total function”. (The important thing here is that the statement is at least Pi2---true pure existence statements are always provable, and if the statements were universal there would be a different ω-consistency problem. The exact statement isn’t so important, but not that these statements are true, but not provable in PA.)
Now consider the statement T=”there is an n such that P(n) is false”. PA+T has no standard model (because T is false), but PA+T doesn’t prove any of the P(n), let alone all of them, so there’s no ω-consistency problem.
Thanks, can you recommend a textbook for this stuff? I’ve mostly been learning off Wikipedia.
I can’t find a textbook on logic in the lesswrong textbook list.
I’m a fan of Enderton’s “A Mathematical Introduction to Logic”. It’s short and very precisely written, which can make it a little difficult to learn from on its own, but together with a general familiarity with the subject and using wikipedia for additional examples elaboration, it should be perfect.