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potato comments on Induction, Deduction, and the Collatz Conjecture: the Decidedly Undecidable Propositions. - Less Wrong
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What I'm really asking is, if some statement turns out to be undecidable for all of our Tarskian truth translation maps to models, does that make that conjecture meaningless, or is undecidable somehow distinct from unverifiable. What is the difference between believing "that conjecture is unverifiable" and believing "that conjecture is undecidable."? Are the expectations/restrictions on experience that those two believes offer identical? If so does that mean that the difference between those two believes is a syntactic issue?
See Making Beliefs Pay Rent :
http://lesswrong.com/lw/i3/making_beliefs_pay_rent_in_anticipated_experiences/
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.
"1+1=giraffe" is meaningless "Godel sentence" is undecidable
"1+1=giraffe" isn't meaningless. It means that if you add one and one, you get a giraffe. Everyone knows that's where giraffes come from.
If there is a difference between undecidable and meaningless, and a statement can be shown to be undecidable, then we need to accept that not every meaningful statement is true or false at least in the case of the natural numbers.
So does that mean that we should reject the principle of excluded middle? If so, that means that our standard logics are useless for dealing with mathematical (if not all) reasoning. Intuitionistic logic might be better suited at dealing with these sorts of issues, but it seems strange that some meaningful statements might be neither true nor false.