Fedor Pakhomov

• Russian Academy of Sciences

Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $\beta$-models: $T\prec_\beta U$ if every $\beta$-model of $U$ contains a countable coded $\beta$-model of $T$. The restriction of...

For each [Formula: see text], let [Formula: see text] mean “the sentence [Formula: see text] is true in all [Formula: see text]-correct transitive sets.” Assuming Gödel’s axiom [Formula: see text], we prove the following graded variant of Solovay’s completeness theorem: the set of formulas valid under this interpretation is precisely the set of the...

This paper studies logical aspects of the notion of better-quasi-order, which was introduced by C. Nash-Williams [Proc. Cambridge Philos. Soc. 61 (1965), pp. 697–720; Proc. Cambridge Philos. Soc. 64 (1968), pp. 273–290]. A central tool in the theory of better-quasi-orders is the minimal bad array lemma. We show that this lemma is exceptionally stro...

The famous theorem of Higman states that for any well-quasi-order (wqo) $Q$ the embeddability order on finite sequences over $Q$ is also wqo. In his celebrated 1965 paper, Nash-Williams established that the same conclusion holds even for all the transfinite sequences with finite range, thus proving a far reaching generalization of Higman's theorem....

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf{PA}$. This result is commonly known as Feferman's completeness theorem. The purpose of this paper is twofold. On the one hand this is an expository paper, giving two new proofs of Feferman's completeness...

We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on $\mathbb {N}$ obtained through processes such as the ones that yield multiplication f...

It has recently been shown that fairly strong axiom systems such as $\mathsf{ACA}_0$ cannot prove that the antichain with three elements is a better quasi order ($\mathsf{bqo}$). In the present paper, we give a complete characterization of the finite partial orders that are provably $\mathsf{bqo}$ in such axiom systems. The result will also be exte...

We study the Π 3 1 -soundness spectra of theories. Given a recursively enumerable extension T of A C A 0 , O 3 1 ( T ) is defined as the set of all 2-ptykes on which T is correct about well foundedness. This is a measure of how close T is to being Π 3 1 -sound. We carry out a proof-theoretic classification of theories according to O 3 1 ( T ) , as...

This paper studies logical aspects of the notion of better quasi order, which has been introduced by C. Nash-Williams (Mathematical Proceedings of the Cambridge Philosophical Society 1965 & 1968). A central tool in the theory of better quasi orders is the minimal bad array lemma. We show that this lemma is exceptionally strong from the viewpoint of...

We show that there is no theory that is minimal with respect to interpretability among recursively enumerable essentially undecidable theories.

We fix a gap in a proof in our paper Reducing[Formula: see text]-model reflection to iterated syntactic reflection.

We develop the abstract framework for a proof‐theoretic analysis of theories with scope beyond the ordinal numbers, resulting in an analog of ordinal analysis aimed at the study of theorems of complexity Π21$\Pi ^1_2$. This is done by replacing the use of ordinal numbers by particularly uniform, wellfoundedness preserving functors on the category o...

We prove that any linear order definable in the standard model (Z, <, +) of Presburger arithmetic is (Z, <, +)-definably embeddable into the lexicographic ordering on Z^n, for some n.

We show that there is no theory that is minimal with respect to interpretability among recursively enumerable essentially undecidable theories.

In this paper, we prove that no consistent finitely axiomatized theory one‐dimensionally interprets its own extension with predicative comprehension. This constitutes a result with the flavor of the Second Incompleteness Theorem whose formulation is completely arithmetic‐free. Probably the most important novel feature that distinguishes our result...

We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original system of arithmetic. Much stronger systems, however, are obtained by adding either induction axioms or refle...

Fast-growing hierarchies are sequences of functions obtained through various processes similar to the ones that yield multiplication from addition, exponentiation from multiplication, etc. We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. We show that the catego...

jats:p> In mathematical logic there are two seemingly distinct kinds of principles called “reflection principles.” Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this pa...

We develop the abstract framework for a proof-theoretic analysis of theories with scope beyond ordinal numbers, resulting in an analog of Ordinal Analysis aimed at the study of theorems of complexity $\Pi^1_2$. This is done by replacing the use of ordinal numbers by particularly uniform, wellfoundedness preserving functors in the category of linear...

By a previous result of the authors there are no infinite sequences of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$ such that each of them proves $\Pi^1_1$-reflection of the next one. This engenders a well-founded "reflection ranking" of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$. In previous work the authors proved that for any $\Pi^1_1$-sou...

In mathematical logic there are two seemingly distinct kinds of principles called "reflection principles." Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this paper we s...

Presburger arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture...

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the $\Pi ^1_1...

Presburger Arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger Arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture...

We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original system of arithmetic. Much stronger systems, however, are obtained by adding either induction axioms or refle...

By a well-known result of Kotlarski et al. (1981), first-order Peano arithmetic \({{\mathsf {P}}}{{\mathsf {A}}}\) can be conservatively extended to the theory \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and com...

In the paper we introduce a weak set theory $\mathsf{H}_{<\omega}$ . A formalization of arithmetic on finite von Neumann ordinals gives an embedding of arithmetical language into this theory. We show that $\mathsf{H}_{<\omega}$ proves a natural arithmetization of its own Hilbert-style consistency. Unlike some previous examples of theories proving t...

In this paper, we study finitely axiomatizable conservative extensions of a theory U in the case where U is recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively. Consider a finite expansion of the signature of...

We show that the decision problem for the basic system of interpretability logic IL is PSPACE-complete. For this purpose we present an algorithm which uses polynomial space w.r.t. the complexity of a given formula. The existence of such algorithm, together with the previously known PSPACE hardness of the closed fragment of IL, implies PSPACE-comple...

By a well-known result of Kotlarski, Krajewski, and Lachlan (1981), first-order Peano arithmetic $PA$ can be conservatively extended to the theory $CT^{-}[PA]$ of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quan...

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderness phenomenon by studying a coarsening of the consistency strength order, namely, the $\Pi^1_1$ r...

Let $\operatorname{Con}(\mathbf T)\!\restriction\!x$ denote the finite consistency statement "there are no proofs of contradiction in $\mathbf T$ with $\leq x$ symbols". For a large class of natural theories $\mathbf T$, Pudl\'ak has shown that the lengths of the shortest proofs of $\operatorname{Con}(\mathbf T)\!\restriction\!n$ in the theory $\ma...

Let $\operatorname{Con}(\mathbf T)\!\restriction\!x$ denote the finite consistency statement "there are no proofs of contradiction in $\mathbf T$ with $\leq x$ symbols". For a large class of natural theories $\mathbf T$, Pudl\'ak has shown that the lengths of the shortest proofs of $\operatorname{Con}(\mathbf T)\!\restriction\!n$ in the theory $\ma...

In this paper we provide a (negative) solution to a problem posed by Stanis{\l}aw Krajewski. Consider a recursively enumerable theory U and a finite expansion of the signature of U that contains at least one predicate symbol of arity $\ge$ 2. We show that, for any finite extension $\alpha$ of U in the expanded language that is conservative over U,...

We show that the decision problem for the basic system of interpretability logic IL is PSPACE-complete. For this purpose we present an algorithm which uses polynomial space with respect to the complexity of a given formula. The existence of such algorithm, together with the previously known PSPACE hardness of the closed fragment of IL, implies PSPA...

Presburger arithmetic PrA is the true theory of natural numbers with addition. We study interpretations of PrA in itself. We prove that all one-dimensional self-interpretations are definably isomorphic to the identity self-interpretation. In order to prove the result we show that all linear orders that are interpretable in (N,+) are scattered order...

In this paper we present a new proof of Solovay’s theorem on arithmetical completeness of Gödel-Löb provability logic \(\mathsf {GL}\). Originally, completeness of \(\mathsf {GL}\) with respect to interpretation of \(\Box \) as provability in \(\mathsf {PA}\) was proved by Solovay in 1976. The key part of Solovay’s proof was his construction of an...

In this paper we present a new proof of Solovay's theorem on arithmetical completeness of G\"odel-L\"ob provability logic GL. Originally, completeness of GL with respect to interpretation of $\Box$ as provability in PA was proved by R. Solovay in 1976. The key part of Solovay's proof was his construction of an arithmetical evaluation for a given mo...

The polymodal provability logic GLP was introduced by Japaridze in 1986. It is the provability logic of certain chains of provability predicates of increasing strength. Every polymodal logic corresponds to a variety of polymodal algebras. Beklemishev and Visser asked whether the elementary theory of the free GLP-algebra generated by the constants 0...

Полимодальная логика доказуемости $\mathrm{GLP}$ была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории с...

Formalized provability predicates have a long history of consideration in proof theory. It is well-known that formulas that satisfy Hilbert-Bernays-Löb derivability conditions behave similarly to formalized provability predicates, in particular they satisfy formalized Löb's theorem. In the same fashion as the classical arithmetical semantics for th...

The notion of slow provability for Peano Arithmetic ($\mathsf{PA}$) was introduced by S.D. Friedman, M. Rathjen, and A. Weiermann. They studied the slow consistency statement $\mathrm{Con}_{\mathsf{s}}$ that asserts that a contradiction is not slow provable in $\mathsf{PA}$. They showed that the logical strength of $\mathsf{PA}+\mathrm{Con}_{\maths...

Caucal hierarchy is a well-known class of graphs with decidable monadic theories. It were proved by L. Braud and A. Carayol that well-orderings in the hierarchy are the well-orderings with order types less than $\varepsilon_0$. Naturally, every well-ordering from the hierarchy could be considered as a constructive system of ordinal notations. In pr...

There is a polymodal provability logic $GLP$. We consider generalizations of this logic: the logics $GLP_{\alpha}$, where $\alpha$ ranges over linear ordered sets and play the role of the set of indexes of modalities. We consider the varieties of modal algebras that corresponds to the polymodal logics. We prove that the elementary theories of the f...

We consider the constructive ordinal notation system for the ordinal ${\epsilon_0}$ that were introduced by L.D. Beklemishev. There are fragments of this system that are ordinal notation systems for the smaller ordinals ${\omega_n}$ (towers of ${\omega}$-exponentiations of the height $n$). This systems are based on Japaridze's provability logic $\m...

We consider well-known provability logic GLP. We prove that the GLP-provability problem for variable-free polymodal formulas is PSPACE-complete. For a number n, let L^n_0 denote the class of all polymodal variable-free formulas without modalities , <n+1>,... . We show that, for every number n, the GLP-provability problem for formulas from L^n_0 is...

The Lindenbaum algebra of Peano PA can be enriched by the -consistency operators which assign, to a given formula, the statement that the formula is compatible with the theory PA extended by the set of all true -sentences. In the Lindenbaum algebra of PA, a lower semilattice is generated from by the -consistency operators. We prove the undecidabili...