1,108 research outputs found
From Tarski to G\"odel. Or, how to derive the Second Incompleteness Theorem from the Undefinability of Truth without Self-reference
In this paper, we provide a fairly general self-reference-free proof of the
Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of
Truth.Comment: 7 page
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
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
Наші презентації
Кодак М.П. Авторська свідомість і класична поетика. –
К.: ПЦ “Фоліант”, 2006. – 336 с.Людмила Качмар. Іван Теодор Рудницький: життя на тлі
історії / Львівський національний університет ім. І.Франка. –
Л.: Астролябія, 2006. – 216 с.Мельниченко В. Тарас Шевченко: “Моє перебування в Москві”. – М.: ОЛМА
Медиа Групп, 2007. – 512 с.Konstanty romantyzmu а Sergej Макаrа. – Banska Bystrica, 2006. – 322 с.Мусhajlo Najenko. Ukrajinsky literarny romantizmus –
dobody a naddobody. – Banska Bystrica, 2006. – 108 с.“Людмила Тарнашинська”: Науково,допоміжний біобібліогра,
фічний покажчик / Бібліограф-укладач Г.Волянська, наук. ред.
В.Кононенко. – К., 2006. – 118 с. / Міністерство культури і туризму
України та Національна Парламентська бібліотека України.Всесвіт. – 2007. – №5,6.Кур’єр Кривбасу. – 2007. – №210,211 (травень,червень)
The predicative Frege hierarchy
AbstractIn this paper, we characterize the strength of the predicative Frege hierarchy, Pn+1V, introduced by John Burgess in his book [J. Burgess, Fixing frege, in: Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005]. We show that Pn+1V and Q+conn(Q) are mutually interpretable. It follows that PV:=P1V is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [M. Ganea, Burgess’ PV is Robinson’s Q, The Journal of Symbolic Logic 72 (2) (2007) 619–624] using a different proof. Another consequence of the our main result is that P2V is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, IΔ0+EXP, Q3). The fact that P2V interprets EA was proved earlier by Burgess. We provide a different proof.Each of the theories Pn+1V is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, PωV, is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with PωV is finitely axiomatizable
Essential Hereditary Undecidability
In this paper we study \emph{essential hereditary undecidability}. Theories
with this property are a convenient tool to prove undecidability of other
theories. The paper develops the basic facts concerning essentially hereditary
undecidability and provides salient examples, like a construction of \ehu\
theories due to Hanf and an example of a rather natural essentially
hereditarily undecidable theory strictly below {\sf R}. We discuss the
(non-)interaction of essential hereditary undecidability with recursive boolean
isomorphism.
We develop a reduction relation \emph{essential tolerance}, or, in the
converse direction, \emph{lax interpretability} that interacts in a good way
with essential hereditary undecidability.
We introduce the class of -friendly theories and show that
-friendliness is sufficient but not necessary for essential
hereditary undecidability.
Finally, we adapt an argument due to Pakhomov, Murwanashyaka and Visser to
show that there is no interpretability minimal essentially
hereditarilyundecidable theory
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
- …