Saeed Salehi

University of Tabriz · Department of Pure Mathematics

We unify Gödel’s first incompleteness theorem (1931), Tarski’s undefinability theorem (1933), Gödel-Carnap’s diagonal lemma (1934) and Rosser’s (strengthening of Gödel’s first) incompleteness theorem (1936), whose proofs resemble much and use almost the same technique.

We unify Godel's First Incompleteness Theorem (1931), Tarski's Undefinability Theorem (1933), Godel-Carnap's Diagonal Lemma (1934), and Rosser's (strengthening of Godel's first) Incompleteness Theorem (1936), whose proofs resemble much and use almost the same technique.

Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable. These sentences are independent when the theory is so-called omega-consistent; a notion introduced by Godel, whi...

There is a long-standing debate in the logico-philosophical community as to if/why the Gödelian sentences of a consistent and sufficiently strong theory are true ($\gamma $ is a Gödelian sentence of $T$ when $\gamma $ is equivalent to the $T$-unprovability of $\gamma $ inside $T$). The prevalent argument seems to be something like the following: si...

We prove the equivalence of the semantic version of Tarski’s theorem on the undefinability of truth with the semantic version of the diagonal lemma and also show the equivalence of a syntactic version of Tarski’s undefinability theorem with a weak syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for these theorems from the li...

Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Gödel (Kurt Gödel collected works (Volume I) Publications 1929–1936, Oxford University Press, pp 222–225, 1932), and it is proved by Jaśkowski (Actes du Congrés International de Philosophie Scientifique, VI. Philosophie des Mathématiques, Actualités Scientifiques...

We argue that, under the usual assumptions for sufficiently strong arithmetical theories that are subject to Gödel’s First Incompleteness Theorem, one cannot, without impropriety, talk about the Gödel sentence of the theory. The reason is that, without violating the requirements of Gödel’s theorem, there could be a true sentence and a false one eac...

Interesting as they are by themselves in philosophy and mathematics, paradoxes can be made even more fascinating when turned into proofs and theorems. For example, Russell’s paradox, which overthrew Frege’s logical edifice, is now a classical theorem in set theory, to the effect that no set contains all sets. Paradoxes can be used in proofs of some...

There is a longstanding debate in the logico-philosophical community as to why the G\"odelian sentences of a consistent and sufficiently strong theory are true. The prevalent argument seems to be something like this: since every one of the G\"odelian sentences of such a theory is equivalent to the theory's consistency statement, even provably so in...

We prove the equivalence of the semantic version of Tarski's theorem on the undefinability of truth with a semantic version of the Diagonal Lemma, and also show the equivalence of syntactic Tarski's Undefinability Theorem with a weak syntactic diagonal lemma. We outline three seemingly diagonal-free proofs for these theorems from the literature, an...

Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Kurt G\"odel (1932), and it is proved by Stanis{\l}aw Ja\'skowski (1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Saul Kripke (1959). G\"odel's proof gave rise to an intermediate propos...

Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of Complete Ordered Fields with which Real Analysis starts. Groups abound in mathematical sciences, while by Dedekind'...

A cornerstone of modern mathematical logic is the diagonal lemma of Gödel and Carnap. It is used in, for example, the classical proofs of the theorems of Gödel, Rosser, and Tarski. From its first explication in 1934, just essentially one proof has appeared for the diagonal lemma in the literature; a proof that is so tricky and hard to relate that m...

The proofs of Gödel (1931), Rosser (1936), Kleene (first 1936 and second 1950), Chaitin (1970), and Boolos (1989) for the first incompleteness theorem are compared with each other, especially from the viewpoint of the second incompleteness theorem. It is shown that Gödel’s (first incompleteness theorem) and Kleene’s first theorems are equivalent wi...

It is quite well-known from Kurt G¨odel’s (1931) ground-breaking Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are representable in sufficiently strong arithmetical theories. It is also known, though perhaps not as well-known as the forme...

Paradoxes are interesting puzzles in philosophy and mathematics, and they can be even more fascinating when turned into proofs and theorems. For example, Liar's paradox can be translated into a propositional tautology, and Barber's paradox into a first-order tautology. Russell's paradox, which collapsed Frege's foundations for mathematics, is now a...

It is quite well-known from Kurt G\"odel's (1931) ground-breaking result on the Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are representable in sufficiently strong arithmetical theories. It is also known, though perhaps not as well-kno...

Over the last century, the principle of “continuous induction” has been studied by different authors in different formats. All of these different forms are equivalent to one of the three versions that we isolate in this paper. We show that one of the three forms of continuous induction is weaker than the other two by proving that it is equivalent t...

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication,...

The proofs of Kleene, Chaitin and Boolos for Gödel's First Incompleteness Theorem are studied from the perspectives of constructivity and the Rosser property. A proof of the incompleteness theorem has the Rosser property when the independence of the true but unprovable sentence can be shown by assuming only the (simple) consistency of the theory. I...

We take an argument of Gödel's from his ground‐breaking 1931 paper, generalize it, and examine its validity. The argument in question is this: the sentence G says about itself that it is not provable, and G is indeed not provable; therefore, G is true.

The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the co...

Kripke frames (and models) provide a suitable semantics for sub-classical logics; for example Intuitionistic Logic (of Brouwer and Heyting) axiomatizes the reflexive and transitive Kripke frames (with persistent satisfaction relations), and the Basic Logic (of Visser) axiomatizes transitive Kripke frames (with persistent satisfaction relations). He...

A universal schema for diagonalization was popularized by N.S. Yanofsky (2003), based on a pioneering work of F.W. Lawvere (1969), in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function. It was shown that many self-referential paradoxes and diagonally proved theorems can...

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication,...

The multiplicative theory of a set of numbers is the theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the complex, real and (positive) rational numbers. These theories are known to be decid...

Axiomatizing mathematical structures is a goal of Mathematical Logic. Axiomatizability of the theories of some structures have turned out to be quite difficult and challenging, and some remain open. However axiomatization of some mathematical structures are now classical theorems in Logic, Algebra and Geometry. In this paper we will study the axiom...

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of addition and multiplication is undecidable in the domains of N, Z and Q, though Tarski proved that this theory is dec...

The proofs of Chaitin and Boolos for Godel's Incompleteness Theorem are studied from the perspectives of constructibility and Rosserizability. By Rosserization of a proof we mean that the independence of the true but unprovable sentence can be shown by assuming only the (simple) consistency of the theory. It is known that Godel's own proof for his...

Gödel–Rosser's Incompleteness Theorem is generalized by showing Π n + 1 Πn+1-incompleteness of any Σ n + 1 Σn+1-definable extension of Peano Arithmetic which is either Σ n Σn-sound or n n-consistent. The optimality of this result is proved by presenting a complete, Σ n + 1 Σn+1-definable, Σ n − 1 Σn−1-sound, and ( n − 1 ) (n−1)-consistent theory fo...

We take an argument of G\"odel's from his groundbreaking 1931 paper, generalize it, and examine its (in)validity. The argument in question is this: "$A$ says of itself that it has property $F$, and $A$ does have property $F$; therefore, $A$ is true."

We show that the existence of a finitely axiomatized theory which can prove all the true $\Sigma_1$ sentences may imply Godel's Second Incompleteness Theorem, by incorporating some bi-theoretic version of the derivability conditions (first discussed by Detlefsen~2001). We also argue that Tarski's theorem on the undefinability of truth is Godel's fi...

Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a syntactic-semantic notion (that is the consistency of a theory with the set of all true $\Pi_n$ sentences or equivalently the $\Sigma_n$ soundness of the theory) that corresponds to Godel's notion of $\omega$-consi...

We argue that Gödel's completeness theorem is equivalent to completability of consistent theories, and Gödel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursiv...

To counter a general belief that all the paradoxes stem from a kind of circularity (or involve some self--reference, or use a diagonal argument) Stephen Yablo designed a paradox in 1993 that seemingly avoided self--reference. We turn Yablo's paradox, the most challenging paradox in the recent years, into a genuine mathematical theorem in Linear Tem...

A universal scheme for diagonalization was popularized by N. S. Yanofsky (Bull. Symb. Logic 9, 2003, 362--386) in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function. It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that scheme....

It is argued that G\"odel's completeness theorem is equivalent to completability of consistent theories, and G\"odel's incompleteness theorem expresses that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enum...

The problem of Π1–separating the hierarchy of bounded arithmetic has been studied in the article. It is shown that the notion of Herbrand consistency, in its full generality, cannot Π1–separate the theory IΔ0+⋀jΩj from IΔ0; though it can Π1–separate IΔ0+Exp from IΔ0. Namely, we show the unprovability of the Herbrand consistency of IΔ0 in the theory...

We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of IΔ0 whose Herbrand Consistency is not provable in IΔ0. We also show the existence of an IΔ0-derivable Π 1-sentence such that IΔ0 cannot prove its Herbrand Consistency.

Gödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae , vol. 171 (2002), pp. 279–292]. In that paper, it w...

We axiomatize two important intermediate logics, classifying end-extension Kripke models and cofinal-extension Kripke models. As applications, we show that Heyting Arithmetic, HA, is complete with respect to the class of its end-extension Kripke models and every cofinal-extension Kripke model of HA is PA-normal.

We consider several aspects of Wilke's [T. Wilke, An algebraic characterization of frontier testable tree languages, Theoret. Comput. Sci. 154 (1996) 85-106] tree algebra formalism for representing binary labelled trees and compare it with approaches that represent trees as terms in the traditional way. A convergent term rewriting system yields nor...

We consider varieties of recognizable subsets of many-sorted finitely generated free algebras over a given variety, varieties of congruences of such algebras, and varieties of finite many-sorted algebras. A variety theorem that establishes bijections between the classes of these three types of varieties is proved. For this, appropriate notions of m...

Pin's variety theorem for positive varieties of string languages and varieties of finite ordered semigroups is proved for trees, i.e., a bijective correspondence between positive varieties of tree languages and varieties of finite ordered algebras is established. This, in turn, is extended to generalized varieties of finite ordered algebras, which...

A polynomially bounded recursive realizability, in which the recursive functions used in Kleene's realizability are restricted to polynomially bounded functions, is introduced. It is used to show that provably total functions of Ruitenburg's Basic Arithmetic are polynomially bounded (primitive) recursive functions. This sharpens our earlier result...

As a framework for characterizing families of regular languages of binary trees, Wilke introduced a formalism for defining binary trees that uses six many-sorted operations involving letters, trees and contexts. In this paper a completeness property of these operations is studied. It is shown that all functions involving letters, binary trees and b...

Diss. -- Turun yliopisto.

We present axiom systems, and provide soundness and strong completeness theorems, for classes of Kripke models with restricted extension rules among the node structures of the model. As examples we present an axiom system for the class of cofinal extension Kripke models, and an axiom system for the class of end-extension Kripke models. We also show...

It is shown that all the provably total functions of Basic Arithmetic BA, a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which BA is sound, is introduced. This realizability is similar to Kleene's recursive realizability, except that re...

Wilke’s tree algebra formalism for characterizing families of tree languages is based on six operations involving letters, binary trees and binary contexts. In this paper a completeness property of these operations is studied. It is claimed that all functions involving letters, binary trees and binary contexts which preserve all syntactic tree alge...

By formalizing some classical facts about provably total functions of intuitionistic primitive recursive arithmetic (iPRA), we prove that the set of decidable formulas of iPRA and of iΣ 1 + (intuitionistic Σ 1 -induction in the language of PRA) coincides with the set of its provably Δ 1 -formulas and coincides with the set of its provably atomic fo...