I think you're reading too much into what I'm saying. I'm not suggesting that second order arithmetic is useful as a mathematical framework to talk about reasoning, in the way that first-order logic can. I'm saying that second order arithmetic is a useful way to talk about what makes the natural numbers special.
I'm also not suggesting that second order arithmetic has anything deep to add relative to a naïve (but sufficiently abstract) understanding of induction, but given that many people don't have a sufficiently abstract understanding of induction, I t...
I think this critique undervalues the insight second order arithmetic gives for understanding the natural numbers.
If what you wanted from logic was a complete, self-contained account of mathematical reasoning then second order arithmetic is a punt: it purports to describe a set of rules for describing the natural numbers, and then in the middle says "Just fill in whatever you want here, and then go on". Landsburg worries that, as a consequence, we haven't gotten anywhere (or, worse, have started behind where we started, since no we need an accou...
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 funct...
ω-inconsistency isn't exactly the same thing as being false in the standard model. Being ω-inconsistent requires both that the theory prove all the statements P(n) for standard natural numbers n, but also prove that there is an n for which P(n) fails. Therefore a theory could be ω-consistent because it fails to prove P(n), even though P(n) is true in the standard model. So even if we could check ω-consistency, we could take PA, add an axiom T, and end up with an ω-consistent theory which nonetheless is not true in the standard model.
By the way, there ar...
Everyone seems to be taking the phrase "human Gödel sentence" (and, for that matter, "the Gödel sentence of a turing machine") as if its widely understood, so perhaps it's a piece of jargon I'm not familiar with. I know what the Gödel sentence of a computably enumerable theory is, which is the usual formulation. And I know how to get from a computably enumerable theory to the Turing machine which outputs the statements of that theory. But not every Turing machine is of this form, so I don't know what it means to talk about the Gödel ...
Took the survey. I just missed last year's, so I was glad to get to participate this year.