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ZFC is probably the biggest foundation, and only Choice is apparently controversial. The results aren't that weird, they're just different and occasionally more
by ajs1998 28d ago
ZFC is probably the biggest foundation, and only Choice is apparently controversial. The results aren't that weird, they're just different and occasionally more useful than using !Choice.
- andriy_koval 28d agodo we know if claude's formalization is built on top of zfc and not zfc+extra? zfc itself is not sufficient, you need some layers of extra concepts formalization to fit specific problem domain(e.g. zfc doesn't define even basic arithmetics), which also could have potential issues.
- drdeca 28d agoWithin a given inference system, one can define concepts. This doesn’t add any axioms. It is, in essence, just a way to abbreviate things.
- andriy_koval 28d agook, you now added some unknown inference system in addition to zfc
- drdeca 27d agoNo, it is the same inference system. They are just abbreviations.
- andriy_koval 27d agoand what is that system?
- drdeca 26d agoZFC
- andriy_koval 26d agozfc is a bunch of axioms and not inference system. It is commonly assumed that it is built on top of some unspecified first order logic which commonly assumed to include bunch of inference rules. There is no ground truth in my understanding where this all is formally defined.
- drdeca 26d agoIf you want to use “ZFC” to refer to the axioms without any rules of inference, I guess you can do that, but when someone refers to “ZFC” when they are filling a slot that needs a (axioms + rules of inference), the obvious interpretation is that they are referring to the usual system of ZFC.
- andriy_koval 26d ago> when someone refers to “ZFC” when they are filling a slot that needs a (axioms + rules of inference), the obvious interpretation is that they are referring to the usual system of ZFC. its bro-math. In formal math you need to be specific what inference system you use. There are many of them. Then you need to have formal proof that in that system you can derive concept of function and then think about question if it won't make paradoxes and contradictions with ZFC.
- Smaug123 27d agoClaude’s formalisation, being in Lean, is based on the calculus of inductive constructions, not ZFC. In Lean 3, per Carneiro, any theorem of Lean 3’s theory can be proved in ZFC plus some finite number of inaccessible cardinals (and, IIRC, vice versa). The precise strength of Lean 4 is not quite clear yet, I think (I guess this is partly what Lean4Lean is hoping to address).
- lanstin 28d agoReply to sibling - lean4 doesn't rest on ZF or ZFC. https://lean-lang.org/theorem_proving_in_lean4/Axioms-and-Computation/ https://lean-lang.org/theorem_proving_in_lean4/Axioms-and-Co... However I believe an equivalence of power has been shown between the two.
- mietek 28d agoRoughly, yes. See B. Werner (1997) “Sets in types, types in sets”.