arXiv subjects

Saeed Salehi

Publications and source records attributed to Saeed Salehi.

36 records · Page 2Linked to original sources

arXiv2020

Sometime a Paradox, Now Proof: Non-First-Order-izability of Yablo's Paradox

Paradoxes are interesting puzzles in philosophy and mathematics, and they could 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 turns into a first-order tautology. Russell's paradox, which collapsed Frege's foundational framework, is now a classical theorem in set theory, implying that no set of all sets can exist. Paradoxes can be used in proofs of some other theorems; Liar's paradox has been used in the classical proof of Tarski's theorem on the undefinability of truth in sufficiently rich languages. This paradox (and also Richard's paradox) appears implicitly in Gödel's proof of his celebrated first incompleteness theorem. In this paper, we study Yablo's paradox from the viewpoint of first and second order logics. We prove that a formalization of Yablo's paradox (which is second-order in nature) is non-first-order-izable in the sense of George Boolos (1984).

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arXiv2020

On the Notions of Rudimentarity, Primitive Recursivity and Representability of Functions and Relations

It is quite well-known from Kurt Godel'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-known as the former one, that some primitive recursive relations are not rudimentary. We present a simple and elementary proof of this fact in the first part of the paper. In the second part, we review some possible notions of representability of functions studied in the literature, and give a new proof of the equivalence of the weak representability with the (strong) representability of functions in sufficiently strong arithmetical theories. Our results shed some new light on the notions of rudimentary, primitive recursive, and representable functions and relations, and clarify, hopefully, some misunderstandings and confusing errors in the literature.

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arXiv2019

Theorems of Tarski's Undefinability and Godel's Second Incompleteness-Computationally

We present a version of Gödel's Second Incompleteness Theorem for recursively enumerable consistent extensions of a fixed axiomatizable theory, by incorporating some bi-theoretic version of the derivability conditions. We also argue that Tarski's theorem on the Undefinability of Truth is Gödel's First Incompleteness Theorem relativized to definable oracles; a unification of these two theorems is given.

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arXiv2019

First-Order Continuous Induction, and a Logical Study of Real Closed Fields

Over the last century, the principle of "induction on the continuum" has been studied by different authors in different formats. All of these different readings are equivalent to one of the three versions that we isolate in this paper. We also formalize those three forms (of "continuous induction") in first-order logic and prove that two of them are equivalent and sufficiently strong to completely axiomatize the first-order theory of the real closed (ordered) fields. We show that the third weaker form of continuous induction is equivalent with the Archimedean property. We study some equivalent axiomatizations for the theory of real closed fields and propose a first-order scheme of the fundamental theorem of algebra as an alternative axiomatization for this theory (over the theory of ordered fields).

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arXiv2018

On Decidability of the Ordered Structures of Numbers

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, it is known that the theories of $\mathbb{N}$ and $\mathbb{Z}$ are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski's theorem, the multiplicative ordered structure of $\mathbb{R}$ is decidable also; here we prove this result directly and present an axiomatization. The structure of $\mathbb{Q}$ in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure we prove that it is not finitely axiomatizable.

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arXiv2018

On Constructivity and the Rosser Property: a closer look at some Gödelean proofs

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. It is known that Gödel's own proof for his incompleteness theorem does not have the Rosser property, and we show that neither do Kleene's or Boolos' proofs. However, we show that a variant of Chaitin's proof can have the Rosser property. The proofs of Gödel, Rosser and Kleene are constructive in the sense that they explicitly construct, by algorithmic ways, the independent sentence(s) from the theory. We show that the proofs of Chaitin and Boolos are not constructive, and they prove only the mere existence of the independent sentences.

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arXiv2018

On Arithmetical Truth of the Self-Referential Sentences

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.

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arXiv2018

On Axiomatizability of the Multiplicative Theory of Numbers

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 complex, real and (positive) rational numbers. These theories (and also the multiplicative theories of natural and integer numbers) are known to be decidable (i.e., there exists an algorithm that decides whether a given sentence is derivable form the theory); here we present explicit axiomatizations for them and show that they are not finitely axiomatizable. For each of these sets (of complex, real and [positive] rational numbers) a language, including the multiplication operation, is introduced in a way that it allows quantifier elimination (for the theory of that set).

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arXiv2016

Axiomatizing Mathematical Theories: Multiplication

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 axiomatizability of the theories of multiplication in the domains of natural, integer, rational, real, and complex numbers. We will review some classical theorems, and will give some new proofs for old results. We will see that some structures are missing in the literature, thus leaving it open whether the theories of that structures are axiomatizable (decidable) or not. We will answer one of those open questions in this paper.

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arXiv2016

Computation in Logic and Logic in Computation

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 decidable in the domains of R and C. The theory of multiplication and order (x,<) behaves differently in the above mentioned domains of numbers. By a theorem of Robinson, addition is definable by multiplication and order in the domain of natural numbers, thus the theory (N;x,<) is undecidable. By a classical theorem in mathematical logic, addition is not definable in terms of multiplication and order in R. In this paper, we extend Robinson's theorem to the domain of integers (Z) by showing the definability of addition in (Z;x,<), this implies that (Z;x,<) is undecidable. We also show the decidability of (Q;x,<) by the method of quantifier elimination. Whence, addition is not definable in (Q;x,<).

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arXiv2016

Diagonalizing by Fixed-Points

A universal schema for diagonalization was popularized by N. S. Yanofsky (2003) 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 schema. Here, we fit more theorems in the universal schema of diagonalization, such as Euclid's theorem on the infinitude of the primes and new proofs of Boolos (1997) for Cantor's theorem on the non-equinumerosity of a set with its powerset. Then, in Linear Temporal Logic, we show the non-existence of a fixed-point in this logic whose proof resembles the argument of Yablo's paradox. Thus, Yablo's paradox turns for the first time into a genuine mathematico-logical theorem in the framework of Linear Temporal Logic. Again the diagonal schema of the paper is used in this proof, and also it is shown that G. Priest's inclosure schema (1997) can fit in our universal diagonal/fixed-point schema. We also show the existence of dominating (Ackermann-like) functions (which dominate a given countable set of functions---like primitive recursives) using the schema.

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arXiv2016

Herbrand Consistency of Some Finite Fragments of Bounded Arithmetical Theories

We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of ${\rm IΔ_0}$ whose Herbrand Consistency is not provable in the thoery ${\rm IΔ_0}$. We also show the existence of an ${\rm IΔ_0}-$derivable $Π_1-$sentence such that ${\rm IΔ_0}$ cannot prove its Herbrand Consistency.

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arXiv2016

Kripke Semantics for Fuzzy Logics

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). Here, we investigate whether Kripke frames/models could provide a semantics for fuzzy logics. For each axiom of the Basic Fuzzy Logic, necessary and sufficient conditions are sought for Kripke frames/models which satisfy them. It turns out that the only fuzzy logics (logics containing the Basic Fuzzy Logic) which are sound and complete with respect to a class of Kripke frames/models are the extensions of the Gödel Logic (or the super-intuitionistic logic of Dummett), indeed this logic is sound and strongly complete with respect to reflexive, transitive and connected (linear) Kripke frames (with persistent satisfaction relations). This provides a semantic characterization for the Gödel Logic among (propositional) fuzzy logics.

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arXiv2016

Separating Bounded Arithmetics by Herbrand Consistency

The problem of $Π_1-$separating the hierarchy of bounded arithmetic has been studied in the paper. It is shown that the notion of Herbrand Consistency, in its full generality, cannot $Π_1-$separate the theory ${\rm IΔ_0+\bigwedge_jΩ_j}$ from ${\rm IΔ_0}$; though it can $Π_1-$separate ${\rm IΔ_0+Exp}$ from ${\rm IΔ_0}$. This extends a result of L. A. Kołodziejczyk (2006), by showing the unprovability of the Herbrand Consistency of ${\rm IΔ_0}$ in the theory ${\rm IΔ_0+\bigwedge_jΩ_j}$.

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arXiv2016

Godel-Rosser's Incompleteness Theorems for Non-Recursively Enumerable Theories

Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true $Π_n$-sentences or equivalently the $Σ_n$-soundness of the theory, and the other is $n$-consistency the restriction of $ω$-consistency to the $Σ_n$-formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in a sense. Though the proof of the incompleteness theorem using the $Σ_n$-soundness assumption is constructive, it is shown that there is no constructive proof for the incompleteness theorem using the $n$-consistency assumption, for $n\!>\!2$.

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arXiv2016

Herbrand Consistency of Some Arithmetical Theories

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, \textit{Fundamenta Mathematicae} 171 (2002) 279--292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories ${\rm IΔ_0+Ω_m}$ with $m\geqslant 2$, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory $T\supseteq {\rm IΔ_0+Ω_2}$ in $T$ itself. In this paper, the above results are generalized for ${\rm IΔ_0+Ω_1}$. Also after tailoring the definition of Herbrand consistency for ${\rm IΔ_0}$ we prove the corresponding theorems for ${\rm IΔ_0}$. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories ${\rm IΔ_0+Ω_1}$ and ${\rm IΔ_0}$.

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arXiv2016

Godel's Incompleteness Phenomenon - Computationally

We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel'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 recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of such theories are undecidable.

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arXiv2014

Theoremizing Yablo's Paradox

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 Temporal Logic (LTL). Indeed, Yablo's paradox comes in several varieties; and he showed in 2004 that there are other versions that are equally paradoxical. Formalizing these versions of Yablo's paradox, we prove some theorems in LTL. This is the first time that Yablo's paradox(es) become new(ly discovered) theorems in mathematics and logic.

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