5 research outputs found
Independent Combinatoric Worm Principles for First Order Arithmetic and Beyond
Treballs Finals del MĂ ster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2019-2020, Tutor: Joost J. JoostenIn this thesis we study Beklemishevâs combinatorial principle Every Worm Dies, EWD which although true, it is unprovable in Peano Arithmetic (PA). The principle talks about sequences of modal formulas, the finiteness of all of them being equivalent to the one-consistency of PA. We present the elements of proof theory at play here and perform two attempts at generalizing this theorem. One is directed towards its relationship with some known fragments of PA while the other aims to see its connection with fragments of second order arithmetic
The universal tangle for spatial reasoning
The topological -calculus has gathered attention in recent years as a
powerful framework for representation of spatial knowledge. In particular,
spatial relations can be represented over finite structures in the guise of
weakly transitive wK4 frames. In this paper we show that the topological
-calculus is equivalent to a simple fragment based on a variant of the
`tangle' operator. Similar results were proven for transitive frames by Dawar
and Otto, using modal characterisation theorems for the corresponding classes
of frames. However, since these theorems are not available in our setting,
which has the upshot of providing a more explicit translation and upper bounds
on formula size.Comment: 20 page
Hyperarithmetical Worm Battles
Japaridzeâs provability logic GLP has one modality [n] for each natural number and has been used by Beklemishev for a proof theoretic analysis of Peano arithmetic (PA) and related theories. Among other benefits, this analysis yields the so-called Every Worm Dies (EWD) principle, a natural combinatorial statement independent of PA. Recently, Beklemishev and Pakhomov have studied notions of provability
corresponding to transfinite modalities in GLP. We show that indeed the natural transfinite extension of GLP is sound for this interpretation, and yields independent combinatorial principles for the second order theory ACA of arithmetical comprehension with full induction. We also provide restricted versions of EWD related to the fragments IÎŁn of Peano arithmetic
Arithmetical and Hyperarithmetical Worm Battles
Japaridzeâs provability logic (GLP) has one modality [n] for each natural number and has been used by Beklemishev for a proof theoretic analysis of Peano arithmetic (PA) and related theories. Among other benefits, this analysis yields the so-called Every Worm Dies (EWD) principle, a natural combinatorial statement independent of PA. Recently, Beklemishev and Pakhomov have studied notions of provability corresponding to transfinite modalities in GLP. We show that indeed the natural transfinite extension of GLP is sound for this interpretation and yields independent combinatorial principles for the second-order theory ACA of arithmetical comprehension with full induction. We also provide restricted versions of EWD related to the fragments IÎŁ_n of PA. In order to prove the latter, we show that standard Hardy functions majorize their variants based on tree ordinals