Yes, that is one example of many arbitrary choices in infinite systems. In practice that means that you give up the ability to communicate to anyone else which system you’re working with, unless you also have a channel with which to communicate an infinite amount of information to them.
However the somewhat arbitrary choices I was talking about with respect to infinitary logics was about what rules of inference you use to derive new infinite sentences. Even in finitary logic there are choices such as whether to accept proofs that use Law of Excluded Middle, as well as more esoteric principles such as modal logic, relevance logics, and paraconsistency, but when you have sentences with infinitely many clauses (or even more complex structures) then we need rules that aren’t determined by what happens with every finite number of clauses. Some of these might be very counterintuitive when building from an experience with only finitely many terms, but we can’t say they’re wrong.
Yes, that is one example of many arbitrary choices in infinite systems. In practice that means that you give up the ability to communicate to anyone else which system you’re working with, unless you also have a channel with which to communicate an infinite amount of information to them.
Well, that’s trivially simulatable, since I can generate an infinitely large communication channel, so that’s not a constraint.
Yes, that is one example of many arbitrary choices in infinite systems. In practice that means that you give up the ability to communicate to anyone else which system you’re working with, unless you also have a channel with which to communicate an infinite amount of information to them.
However the somewhat arbitrary choices I was talking about with respect to infinitary logics was about what rules of inference you use to derive new infinite sentences. Even in finitary logic there are choices such as whether to accept proofs that use Law of Excluded Middle, as well as more esoteric principles such as modal logic, relevance logics, and paraconsistency, but when you have sentences with infinitely many clauses (or even more complex structures) then we need rules that aren’t determined by what happens with every finite number of clauses. Some of these might be very counterintuitive when building from an experience with only finitely many terms, but we can’t say they’re wrong.
Well, that’s trivially simulatable, since I can generate an infinitely large communication channel, so that’s not a constraint.