As I understand it, that point about “somewhat arbitrary choices in how finite logic should be extended to infinitary” would also include, for every one of the infinitely many undecidable-by-a-non-hypercomputer propositions, a free choice of whether to include a proposition or its negation as an axiom. Well, almost. Each freely chosen axiom has infinitely many consequences that are the no longer free choices. But it’s still (countably) infinitely many choices. But if you have countably infinitely many binary choices, that’s like choosing a random real number between 0 and 1, so there are uncountably many ways of making that extension, each of which results in a distinct infinitary system. Your proposed hypercomputer can generate all of these.
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.
As I understand it, that point about “somewhat arbitrary choices in how finite logic should be extended to infinitary” would also include, for every one of the infinitely many undecidable-by-a-non-hypercomputer propositions, a free choice of whether to include a proposition or its negation as an axiom. Well, almost. Each freely chosen axiom has infinitely many consequences that are the no longer free choices. But it’s still (countably) infinitely many choices. But if you have countably infinitely many binary choices, that’s like choosing a random real number between 0 and 1, so there are uncountably many ways of making that extension, each of which results in a distinct infinitary system. Your proposed hypercomputer can generate all of these.
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.