Hilberts tenth problem, Godels incompleteness, Halting problem, a unifying perspective

We formulate a property $P$ on a class of relations on the natural numbers, and formulate a general theorem on $P$, from which we get as corollaries the insolvability of Hilberts tenth problem, Godels incompleteness theorem, and Turings halting problem. By slightly strengthening the property $P$, we get Tarskis definability theorem, namely that truth is not first order definable. The property $P$ together with a Cantors diagonalization process emphasizes that all the above theorems are a variation on a theme, that of self reference and diagonalization combined. We relate our results to self referential paradoxes, including a formalisation of the Liar paradox, and fixed point theorems. We also discuss the property $P$ for arbitrary rings. We give a survey on Hilberts tenth problem for quadratic rings and for the rationals pointing the way to ongoing research in main stream mathematics involving recursion theory, definability in model theory, algebraic geometry and number theory.

Some model-theoretic results on the 3-valued paraconsistent first-order logic QCiore

In this paper the 3-valued paraconsistent first-order logic QCiore is studied from the point of view of Model Theory. The semantics for QCiore is given by partial structures, which are first-order structures in which each n-ary predicate R is interpreted as a triple of paiwise disjoint sets of n-uples representing, respectively, the set of tuples which actually belong to R, the set of tuples which actually do not belong to R, and the set of tuples whose status is dubious or contradictory. Partial structures were proposed in 1986 by I. Mikenberg, N. da Costa and R. Chuaqui for the theory of quasi-truth (or pragmatic truth). In 2014, partial structures were studied by M. Coniglio and L. Silvestrini for a 3-valued paraconsistent first-order logic called LPT1, whose 3-valued propositional fragment is equivalent to da Costa-DOtavianos logic J3. This approach is adapted in this paper to QCiore, and some important results of classical Model Theory such as Robinsons joint consistency theorem, amalgamation and interpolation are obtained. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order logics.

First order logic properly displayed

We introduce a proper display calculus for first-order logic, of which we prove soundness, completeness, conservativity, subformula property and cut elimination via a Belnap-style metatheorem. All inference rules are closed under uniform substitution and are without side conditions.

First order logic without equality on relativized semantics

Let $alphageq 2$ be any ordinal. We consider the class $mathsf{Drs}_{alpha}$ of relativized diagonal free set algebras of dimension $alpha$. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of $mathsf{Drs}_{alpha}$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class $mathsf{Drs}_{alpha}$ corresponds to first order logic, without equality symbol, with $alpha$-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.

Relevant First-Order Logic $LP^#$ and Currys Paradox resolution

In 1942 Haskell B.Curry presented what is now called Curry paradox which can be found in a logic independently of its stand on negation.In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Currys Paradox and Shaw-Kwei paradox without rejection any contraction postulate is proposed.