What I’m really asking is, if some statement turns out to be undecidable for all of our models,
Nitpick: you don’t mean “models” here, you mean “theories”.
does that make that conjecture meaningless
Why should it?
or is undecidable somehow distinct from unverifiable.
Oh… you’re implicitly assuming a 1920s style verificationism whereby “meaningfulness” = “verifiability”. That’s a very bad idea because most/all statements turn out to be ‘unverifiable’ - certainly all laws of physics.
As for mathematics, the word ‘verifiable’ applied to a mathematical statement simply means ‘provable’ - either that or you’re using the word in a way guaranteed to cause confusion.
Or perhaps by “statement S is verifiable” what you really mean is “there exists an observation statement T such that P(T|S) is not equal to P(T|¬S)”?
Oh and I’m not talking about true or false in terms of provability (necessarily), don’t forget that there is a semantic theory of truth for formal languages called model theory. Falsification or verification of a statement from a language by semantic means works just as well.
You’re right I do mean theory. But importantly I’m including the first order language that holds for the model of the natural numbers. So the interpreted first order language of the natural numbers is included in this usage of “theory”.
I’m using a 1920′s (ish) style of verificationism that considers cases of P(T|S) ≠ P(T|~S) to be cases of a verifiable statement. See Ayler’s Language, Truth, and Logic. The positivists always held that inductively verifiable statements are still verifiable and thus meaningful.
What makes you say that all statements about the laws of physics are unverifiable? If it restricts your expectations for experience, it is a verifiable prediction. Certainly our hypothetical-deductive theories of the natural universe do in fact restrict our expected stimulus, and can be rejected on the grounds that the restrictions are not met.
I’m using the word “verifiable” as it is used in Positivism and Verificationism. A statement S is verifiable if and only if S is a tautology or there is a strong inductive argument with S as the conclusion, which if cogent gives us a probability for S.
Nitpick: you don’t mean “models” here, you mean “theories”.
Why should it?
Oh… you’re implicitly assuming a 1920s style verificationism whereby “meaningfulness” = “verifiability”. That’s a very bad idea because most/all statements turn out to be ‘unverifiable’ - certainly all laws of physics.
As for mathematics, the word ‘verifiable’ applied to a mathematical statement simply means ‘provable’ - either that or you’re using the word in a way guaranteed to cause confusion.
Or perhaps by “statement S is verifiable” what you really mean is “there exists an observation statement T such that P(T|S) is not equal to P(T|¬S)”?
Oh and I’m not talking about true or false in terms of provability (necessarily), don’t forget that there is a semantic theory of truth for formal languages called model theory. Falsification or verification of a statement from a language by semantic means works just as well.
You’re right I do mean theory. But importantly I’m including the first order language that holds for the model of the natural numbers. So the interpreted first order language of the natural numbers is included in this usage of “theory”.
I’m using a 1920′s (ish) style of verificationism that considers cases of P(T|S) ≠ P(T|~S) to be cases of a verifiable statement. See Ayler’s Language, Truth, and Logic. The positivists always held that inductively verifiable statements are still verifiable and thus meaningful.
What makes you say that all statements about the laws of physics are unverifiable? If it restricts your expectations for experience, it is a verifiable prediction. Certainly our hypothetical-deductive theories of the natural universe do in fact restrict our expected stimulus, and can be rejected on the grounds that the restrictions are not met.
I’m using the word “verifiable” as it is used in Positivism and Verificationism. A statement S is verifiable if and only if S is a tautology or there is a strong inductive argument with S as the conclusion, which if cogent gives us a probability for S.