Description: Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc ). Specifically, the identity function maps the universe onto its power class. Compare canth that works for sets.
This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru ): ~PV , being a class, cannot contain proper classes, so it is no larger than V , which is why the identity function "succeeds" in being surjective onto ~P _V (see pwv ). See also the remark in ru about NF, in which Cantor's theorem fails for sets that are "too large". This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004) (Proof shortened by BJ, 29-Dec-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | ncanth | ⊢ I : V –onto→ 𝒫 V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ovi | ⊢ I : V –1-1-onto→ V | |
2 | f1ofo | ⊢ ( I : V –1-1-onto→ V → I : V –onto→ V ) | |
3 | 1 2 | ax-mp | ⊢ I : V –onto→ V |
4 | pwv | ⊢ 𝒫 V = V | |
5 | foeq3 | ⊢ ( 𝒫 V = V → ( I : V –onto→ 𝒫 V ↔ I : V –onto→ V ) ) | |
6 | 4 5 | ax-mp | ⊢ ( I : V –onto→ 𝒫 V ↔ I : V –onto→ V ) |
7 | 3 6 | mpbir | ⊢ I : V –onto→ 𝒫 V |