Use of these symbols is weakly discouraged in published mathematical writing, as is the use of logical connectives such as ∧,∨, and ⇒. The sentiment seems to be that you generally shouldn’t use symbols from formal logic unless you are actually writing out formulas within an explicitly established formal logical theory, with explicitly establish rules of syntax and inference.
In those terms, what surprised me was that the authors did not explicitly establish a formal logical theory of sets. (I also expected explicit syntax and inference in the proofs.) Is formal-logical set theory frowned upon as well?
In my experience, “We’re doing naive set theory” means something like, “We’ll assume, without further justification, that no Russell-style paradox applies to any predicate P where we will actually want to write {x : P(x)}. We’ll just assume the existence of a set answering to this description for any P that we need. We know that there are predicates for which this is not allowed, but we’ll just hope that everything works out okay in the cases where we do it.”
The phrase “naive set theory” also connotes a certain cavalierness about whether the elements in one’s sets are themselves constructed out of sets (as in ZF) or whether instead one is working with urelements (objects in sets that are not themselves sets).
Use of these symbols is weakly discouraged in published mathematical writing, as is the use of logical connectives such as ∧,∨, and ⇒. The sentiment seems to be that you generally shouldn’t use symbols from formal logic unless you are actually writing out formulas within an explicitly established formal logical theory, with explicitly establish rules of syntax and inference.
In those terms, what surprised me was that the authors did not explicitly establish a formal logical theory of sets. (I also expected explicit syntax and inference in the proofs.) Is formal-logical set theory frowned upon as well?
As I understand the phrase, it wouldn’t be “naive set theory” if they did that.
Ah, I didn’t know that that “naive” carried the connotation of “non-formal” in this context. This is good to know, thanks.
In my experience, “We’re doing naive set theory” means something like, “We’ll assume, without further justification, that no Russell-style paradox applies to any predicate P where we will actually want to write {x : P(x)}. We’ll just assume the existence of a set answering to this description for any P that we need. We know that there are predicates for which this is not allowed, but we’ll just hope that everything works out okay in the cases where we do it.”
The phrase “naive set theory” also connotes a certain cavalierness about whether the elements in one’s sets are themselves constructed out of sets (as in ZF) or whether instead one is working with urelements (objects in sets that are not themselves sets).