7 papers
Reflection ranks via infinitary derivations
James Walsh
There is no infinite sequence of -sound extensions of each of which proves -reflection of the next. This engenders a well-founded ``reflection rankin…
On the hierarchy of natural theories
James Walsh
It is a well-known empirical phenomenon that natural axiomatic theories are pre-well-ordered by consistency strength. Without a precise mathematical definition of "natural," it is…
Reducing -model reflection to iterated syntactic reflection
Fedor Pakhomov, James Walsh
In mathematical logic there are two seemingly distinct kinds of principles called "reflection principles." Semantic reflection principles assert that if a formula holds in the whol…
Incompleteness and Jump Hierarchies
Patrick Lutz, James Walsh
This paper is an investigation of the relationship between Gödel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of…
A note on the consistency operator
James Walsh
It is a well known empirical observation that natural axiomatic theories are pre-well-ordered by consistency strength. For any natural theory , the next strongest natural theory…
Reflection ranks and ordinal analysis
Fedor Pakhomov, James Walsh
It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respec…