Comment by andriy_koval
15 hours ago
Not just lean, but math foundation itself, I am not strong expert, but my understanding is that there is no fully recognized axiomatic foundation for modern math, all proposals could lead to some weird results.
15 hours ago
Not just lean, but math foundation itself, I am not strong expert, but my understanding is that there is no fully recognized axiomatic foundation for modern math, all proposals could lead to some weird results.
There is, or rather are, fully recognized axiomatic foundations. You are free to choose one you like. Of the most popular ones is ZFC or ZF, but there are others (some lead to the same results some not). The main criteria for popularity is how useful it is. You can even make your own axiomatic where 2+2=5, but it would be useless.
You probably heard about Goedel Incompleteness -- the proof that the the axiomatic itself cannot be proven, like using ZFC to prove ZFC, but that's another topic.
It would be fun to play with this Anthropic/Lean formalization under different axiomatics.
Interestingly, in his ICM 2026 lecture, Terence Tao specifically mentioned that Lean is not based on ZFC.
> Goedel Incompleteness -- the proof that the the axiomatic itself cannot be proven, like using ZFC to prove ZFC, but that's another topic.
Godel theorems are for systems with basic arithmetic, zfc doesn't include arithmetic, thus are not object of Godel theorems.
If you start with "I'm not a strong expert" maybe you should stop continuing saying wrong stuff. What you just wrote is completely wrong.
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ZFC has greater consistency strength than PA.
If we take ZFC (or some other set theory) as our meta theory, we can easily see that the axiom of infinity (of ZFC) gives a set of natural numbers (using the von Neumann encoding), which, when equipped with the successor function, is a model of the natural numbers.
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> Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC.
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
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That is wildly wrong.
Lean is based on Type Theory not ZFC.
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.
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