212 research outputs found
Provability Logic and the Completeness Principle
In this paper, we study the provability logic of intuitionistic theories of
arithmetic that prove their own completeness. We prove a completeness theorem
for theories equipped with two provability predicates and
that prove the schemes and for
. Using this theorem, we determine the logic of fast provability
for a number of intuitionistic theories. Furthermore, we reprove a theorem
previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the
-provability logic of Heyting Arithmetic
Classical System of Martin-Lof's Inductive Definitions is not Equivalent to Cyclic Proofs
A cyclic proof system, called CLKID-omega, gives us another way of
representing inductive definitions and efficient proof search. The 2005 paper
by Brotherston showed that the provability of CLKID-omega includes the
provability of LKID, first order classical logic with inductive definitions in
Martin-L\"of's style, and conjectured the equivalence. The equivalence has been
left an open question since 2011. This paper shows that CLKID-omega and LKID
are indeed not equivalent. This paper considers a statement called 2-Hydra in
these two systems with the first-order language formed by 0, the successor, the
natural number predicate, and a binary predicate symbol used to express
2-Hydra. This paper shows that the 2-Hydra statement is provable in
CLKID-omega, but the statement is not provable in LKID, by constructing some
Henkin model where the statement is false
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