21 research outputs found
Some observations on the FGH theorem
We investigate the Friedman--Goldfarb--Harrington theorem from two
perspectives. Firstly, in the frameworks of classical and modal propositional
logics, we study the forms of sentences whose existence is guaranteed by the
FGH theorem. Secondly, we prove some variations of the FGH theorem with respect
to Rosser provability predicates.Comment: 28 page
The provability logic of all provability predicates
We prove that the provability logic of all provability predicates is exactly
Fitting, Marek, and Truszczy\'nski's pure logic of necessitation .
Moreover, we introduce three extensions , , and
of and investigate the arithmetical semantics of
these logics. In fact, we prove that , , and
are the provability logics of all provability predicates
satisfying the third condition of the derivabiity conditions, all
Rosser's provability predicates, and all Rosser's provability predicates
satisfying , respectively.Comment: 34 page
Incompleteness and undecidability of theories consistent with
We prove the following version of the first incompleteness theorem that
simultaneously strengthens Mostowski's theorem and Vaught's theorem: For any
c.e. family of consistent extensions of Tarski,
Mostowski and Robinson's arithmetic , there exists a sentence
of arithmetic such that and for all , and .Comment: 14 page
Prenex normalization and the hierarchical classification of formulas
Akama et al. [1] introduced a hierarchical classification of first-order
formulas for a hierarchical prenex normal form theorem in semi-classical
arithmetic. In this paper, we give a justification for the hierarchical
classification in a general context of first-order theories. To this end, we
first formalize the standard transformation procedure for prenex normalization.
Then we show that the classes and introduced in
[1] are exactly the classes induced by and respectively via
the transformation procedure in any first-order theory.Comment: 15 page