> Almost all of the thousands of theorems studied by Simpson and his followers over the past four decades have turned out (somewhat mysteriously) to be reducible to one of five systems of logic spanning both sides of the finite-infinite divide. For instance, Ramsey’s theorem for triples (and all ordered sets with more than three elements) was shown in 1972 to belong at the third level up in the hierarchy, which is infinitistic.
> A breakthrough came in 1995, when the British logician David Seetapun, working with Slaman at Berkeley, proved that RT^2_2 is logically weaker than RT^3_2 and thus below the third level in the hierarchy.
> “Since then, many seminal papers regarding RT^2_2 have been published,” said Weiermann — most importantly, a 2012 result by Jiayi Liu (paired with a result by Carl Jockusch from the 1960s) showed that RT^2_2 cannot prove, nor be proved by, the logical system located at the second level in the hierarchy, one rung below RT^3_2.
So: there are five nested axiomatic systems that have been in common use to classify how much a theorem relies on infinite concepts. RT^2_2 is weaker than the third of those, and hard-to-compare with the second of them. (The second system can't prove it, but it also can't prove the second system.)
The new result says that RT^2_2 is reducible to primitive recursive arithmetic, which should mean that PRA is capable of proving anything RT^2_2 can prove. The article mentions that Mysterious Classification Level 2 is also reducible to primitive recursive arithmetic, so, as far as I understand things, PRA was already a system that fell "between the lines" of the five Mysterious Classification Levels (since Mysterious Level 3 is infinitistic and PRA is not).
> Almost all of the thousands of theorems studied by Simpson and his followers over the past four decades have turned out (somewhat mysteriously) to be reducible to one of five systems of logic spanning both sides of the finite-infinite divide. For instance, Ramsey’s theorem for triples (and all ordered sets with more than three elements) was shown in 1972 to belong at the third level up in the hierarchy, which is infinitistic.
> A breakthrough came in 1995, when the British logician David Seetapun, working with Slaman at Berkeley, proved that RT^2_2 is logically weaker than RT^3_2 and thus below the third level in the hierarchy.
> “Since then, many seminal papers regarding RT^2_2 have been published,” said Weiermann — most importantly, a 2012 result by Jiayi Liu (paired with a result by Carl Jockusch from the 1960s) showed that RT^2_2 cannot prove, nor be proved by, the logical system located at the second level in the hierarchy, one rung below RT^3_2.
So: there are five nested axiomatic systems that have been in common use to classify how much a theorem relies on infinite concepts. RT^2_2 is weaker than the third of those, and hard-to-compare with the second of them. (The second system can't prove it, but it also can't prove the second system.)
The new result says that RT^2_2 is reducible to primitive recursive arithmetic, which should mean that PRA is capable of proving anything RT^2_2 can prove. The article mentions that Mysterious Classification Level 2 is also reducible to primitive recursive arithmetic, so, as far as I understand things, PRA was already a system that fell "between the lines" of the five Mysterious Classification Levels (since Mysterious Level 3 is infinitistic and PRA is not).