26 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
Certified -sentences
In this paper, we study the employment of -sentences with
certificate, i.e., -sentences where a number of principles is added
to ensure that the witness is sufficiently number-like. We develop certificates
in some detail and illustrate their use by reproving some classical results and
proving some new ones. An example of such a classical result is Vaught's
theorem of the strong effective inseparability of . We also develop
the new idea of a theory being -sourced. Using this notion
we can transfer a number of salient results from to a variety of
other theories.Comment: 31 page