Often a lot of the important “assumptions” in a theorem are baked into things like the type signature of a particular variable or the definitions of some key terms; in my toy theorem above I give two examples (completeness and lack of time-dependence). You are going to lose some information about what the theorem says when you convert it from math to English; an author’s job is to communicate the “important” parts of the theorem (e.g. the conclusion, any antecedents that the reader may not agree with, implications of the type signature that limit the applicability of the conclusion), which will depend on the audience.
Yep, I agree with all of this.
Converting math into English is a tricky business.
Often, but not in this case. If authors understood the above points and meant to refer to the Complete Class Theorem, they need only have said:
If an agent has complete, transitive preferences, and it does not pursue dominated strategies, then it must be representable as maximizing expected utility.
(And they probably wouldn’t have mentioned Cox, Savage, etc.)
Yup, good point, I think it doesn’t change the conclusion.
I think it does. If the money-pump for transitivity needs Completeness, and Completeness is doubtful, then the money-pump for transitivity is doubtful too.
I think that’s right.
Yep, I agree with all of this.
Often, but not in this case. If authors understood the above points and meant to refer to the Complete Class Theorem, they need only have said:
If an agent has complete, transitive preferences, and it does not pursue dominated strategies, then it must be representable as maximizing expected utility.
(And they probably wouldn’t have mentioned Cox, Savage, etc.)
I think it does. If the money-pump for transitivity needs Completeness, and Completeness is doubtful, then the money-pump for transitivity is doubtful too.
Upon rereading I realize I didn’t state this explicitly, but my conclusion was the following:
Transitivity depending on completeness doesn’t invalidate that conclusion.
Ah I see! Yep, agree with that.