I didn't know and not mentioned in the article is that there are two different programs in reverse mathematics, one relating to classical mathematics, the other relating to complexity questions. Claude says this: not sure about accuracy
Let me clarify the relationship between PV₁ and reverse mathematics systems, as there are actually two different research programs here:
Two Different Programs
1. Classical Reverse Mathematics (over RCA₀)
The standard reverse mathematics program, pioneered by Harvey Friedman and Stephen Simpson, works over second-order arithmetic and studies which set existence axioms are needed to prove theorems of ordinary mathematics. The main systems form the "Big Five":
RCA₀ (Recursive Comprehension Axiom)
WKL₀ (Weak König's Lemma)
ACA₀ (Arithmetic Comprehension Axiom)
ATR₀ (Arithmetic Transfinite Recursion)
Π¹₁-CA₀ (Π¹₁ Comprehension Axiom)
2. Bounded Reverse Mathematics (over PV₁ or similar)
This is a separate program studying computational complexity rather than computability. It analyzes which theorems require which computational resources.
Let me clarify the relationship between PV₁ and reverse mathematics systems, as there are actually two different research programs here: Two Different Programs 1. Classical Reverse Mathematics (over RCA₀) The standard reverse mathematics program, pioneered by Harvey Friedman and Stephen Simpson, works over second-order arithmetic and studies which set existence axioms are needed to prove theorems of ordinary mathematics. The main systems form the "Big Five": RCA₀ (Recursive Comprehension Axiom) WKL₀ (Weak König's Lemma) ACA₀ (Arithmetic Comprehension Axiom) ATR₀ (Arithmetic Transfinite Recursion) Π¹₁-CA₀ (Π¹₁ Comprehension Axiom) 2. Bounded Reverse Mathematics (over PV₁ or similar) This is a separate program studying computational complexity rather than computability. It analyzes which theorems require which computational resources.