Comment by ajs1998
14 hours 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.
14 hours 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.
Reply to sibling - lean4 doesn't rest on ZF or ZFC. https://lean-lang.org/theorem_proving_in_lean4/Axioms-and-Co... However I believe an equivalence of power has been shown between the two.
Roughly, yes. See B. Werner (1997) “Sets in types, types in sets”.
do 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.
Claude’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).
Within a given inference system, one can define concepts. This doesn’t add any axioms. It is, in essence, just a way to abbreviate things.
ok, you now added some unknown inference system in addition to zfc