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We explore the idea of using automatic and similar kind of presentations of structures to deal with the conceptual problem of natural proof-theoretic ordinal notations. We conclude that this approach still does not meet the goals., Comment: 19 pages

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. In the present paper we show that Nash-Williams' Theorem is provable in the system $\mathsf{ATR}_0$ of second-order arithmetic, thus solving an open problem by Antonio Montalb\'an and proving the reverse-mathematical equivalence of Nash-Williams' Theorem and $\mathsf{ATR}_0$. In order to accomplish this, we establish equivalent characterization of transfinite Higman's order and an order on the cumulative hierarchy with urelements from the starting wqo $Q$, and find some new connection that can be of purely order-theoretic interest. Moreover, in this paper we present a new setup that allows us to develop the theory of $\alpha$-wqo's in a way that is formalizable within primitive-recursive set theory with urelements, in a smooth and code-free fashion.

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 theorem that, we hope, shed light on this mysterious and often overlooked result. On the other hand, we combine one of our proofs with results from computable structure theory due to Ash and Knight to give sharp bounds on the order types of well-orders necessary to attain the completeness for levels of the arithmetical hierarchy.

For each $n\in\mathbb{N}$, let $[n]\phi$ mean "the sentence $\phi$ is true in all $\Sigma_{n+1}$-correct transitive sets." Assuming G\"odel's axiom $V = L$, we prove the following graded variant of Solovay's completeness theorem: the set of formulas valid under this interpretation is precisely the set of theorems of the linear provability logic GLP.3. We also show that this result is not provable in ZFC, so the hypothesis V = L cannot be removed. As part of the proof, we derive (in ZFC) the following purely modal-logical results which are of independent interest: the logic GLP.3 coincides with the logic of closed substitutions of GLP, and is the maximal non-degenerate, normal extension of GLP., Comment: 20 pages, 4 figures

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 extended to infinite orders. As an application, we derive that a version of the minimal bad array lemma is weak over $\mathsf{ACA_0}$. In sharp contrast, a recent result shows that the same version is equivalent to $\Pi^1_2$-comprehension over the stronger base theory $\mathsf{ATR}_0$.

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 reverse mathematics, a framework from mathematical logic. Specifically, it is equivalent to the set existence principle of $\Pi^1_2$-comprehension, over the base theory $\mathsf{ATR_0}$.

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., Comment: 15 pages. To be submitted to Proc. Am. Math. Soc

We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model., Comment: 10 pages

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

Erickson defined the fusible numbers as a set $\mathcal F$ of reals generated by repeated application of the function $\frac{x+y+1}{2}$. Erickson, Nivasch, and Xu showed that $\mathcal F$ is well ordered, with order type $\varepsilon_0$. They also investigated a recursively defined function $M\colon \mathbb{R}\to\mathbb{R}$. They showed that the set of points of discontinuity of $M$ is a subset of $\mathcal F$ of order type $\varepsilon_0$. They also showed that, although $M$ is a total function on $\mathbb R$, the fact that the restriction of $M$ to $\mathbb{Q}$ is total is not provable in first-order Peano arithmetic $\mathsf{PA}$. In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets $\mathcal F$ of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function $g:\mathbb R^n\to\mathbb R$. The most straightforward generalization of $\frac{x+y+1}{2}$ to an $n$-ary function is the function $\frac{x_1+\cdots+x_n+1}{n}$. We show that this function generates a set $\mathcal F_n$ whose order type is just $\varphi_{n-1}(0)$. For this, we develop recursively defined functions $M_n\colon \mathbb{R}\to\mathbb{R}$ naturally generalizing the function $M$. Furthermore, we prove that for any linear function $g:\mathbb R^n\to\mathbb R$, the order type of the resulting $\mathcal F$ is at most $\varphi_{n-1}(0)$. Finally, we show that there do exist continuous functions $g:\mathbb R^n\to\mathbb R$ for which the order types of the resulting sets $\mathcal F$ approach the small Veblen ordinal., Comment: Minor corrections. 26 pages, 1 figure

Japaridze's provability logic $GLP$ has one modality $[n]$ for each natural number and has been used by Beklemishev for a proof theoretic analysis of Peano aritmetic $(PA)$ and related theories. Among other benefits, this analysis yields the so-called Every Worm Dies $(EWD)$ principle, a natural combinatorial statement independent of $PA$. Recently, Beklemishev and Pakhomov have studied notions of provability corresponding to transfinite modalities in $GLP$. We show that indeed the natural transfinite extension of $GLP$ is sound for this interpretation, and yields independent combinatorial principles for the second order theory $ACA$ of arithmetical comprehension with full induction. We also provide restricted versions of $EWD$ related to the fragments $I\Sigma_n$ of Peano arithmetic. In order to prove the latter, we show that standard Hardy functions majorize their variants based on tree ordinals., Comment: 24 pages. Additions have been made for a proof of the equivalence on the variants corresponding to the fragments of $PA$

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 orders. Generalizing the notion of a proof-theoretic ordinal, we define the functorial $\Pi^1_2$ norm of a theory and prove its existence and uniqueness for $\Pi^1_2$-sound theories. From this, we further abstract a definition of the $\Sigma^1_2$- and $\Pi^1_2$-soundness ordinals of a theory; these quantify, respectively, the maximum strength of true $\Sigma^1_2$ theorems and minimum strength of false $\Pi^1_2$ theorems of a given theory. We study these ordinals, developing a proof-theoretic classification theory for recursively enumerable extensions of $\mathsf{ACA}_0$ Using techniques from infinitary and categorical proof theory, generalized recursion theory, constructibility, and forcing, we prove that an admissible ordinal is the $\Pi^1_2$-soundness ordinal of some recursively enumerable extension of $\mathsf{ACA}_0$ if and only if it is not parameter-free $\Sigma^1_1$-reflecting. We show that the $\Sigma^1_2$-soundness ordinal of $\mathsf{ACA}_0$ is $\omega_1^{ck}$ and characterize the $\Sigma^1_2$-soundness ordinals of recursively enumerable, $\Sigma^1_2$-sound extensions of $\Pi^1_1{-}\mathsf{CA}_0$., Comment: 26 pages, 2 figures

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 from the previous results of this kind is that it is applicable to arbitrary weak theories, rather than to extensions of some base theory. The methods used in the proof of the main result yield a new perspective on the notion of sequential theory, in the setting of forcing-interpretations., Comment: 13 pages

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 study connections between these two kinds of reflection principles in the setting of second-order arithmetic. We prove that, for a large swathe of theories, $\omega$-model reflection is equivalent to the claim that arbitrary iterations of uniform $\Pi^1_1$ reflection along countable well-orderings are $\Pi^1_1$-sound. This result yields uniform ordinal analyses of theories with strength between $\mathsf{ACA}_0$ and $\mathsf{ATR}$. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory $T$, which is the operator on countable ordinals that maps the order-type of $\prec$ to the proof-theoretic ordinal of $T+\mathsf{WO}(\prec)$. We obtain precise results about the growth of proof-theoretic dilators as a function of provable $\omega$-model reflection. This approach enables us to simultaneously obtain not only $\Pi^0_1$, $\Pi^0_2$, and $\Pi^1_1$ ordinals but also reverse-mathematical theorems for well-ordering principles., Comment: The proof of Lemma 5.2. in V1 contains a gap that is now fixed. Lemma 2.9 from V1 has been split into multiple claims

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 by A. Visser. In order to prove the result we show that all linear orderings that are interpretable in $(\mathbb{N};+)$ are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations., Comment: Submitted to the JLC. arXiv admin note: text overlap with arXiv:1709.07341

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 reflection axioms on top of them. Theories of this kind can interpret some well-known predicatively reducible fragments of second-order arithmetic such as iterated arithmetical comprehension. We obtain sharp results on the proof-theoretic strength of these systems using methods of provability logic. Reflection principles naturally define unary operators acting on the semilattice of axiomatizable extensions of our basic theory of iterated truth. The substructure generated by the top element of this algebra provides a canonical ordinal notation system for the class of theories under investigation. Using these notations we obtain conservativity relationships for iterated reflection principles of different logical complexity levels corresponding to the levels of the hyperarithmetical hierarchy, i.e., the analogs of Schmerl's formulas. These relationships, in turn, provide proof-theoretic analysis of our systems and of some related predicatively reducible theories. In particular, we uniformly calculate the ordinals characterizing the standard measures of their proof-theoretic strength, such as provable well-orderings, classes of provably recursive functions, and $\Pi_1^0$-ordinals., Comment: 49 pages

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 their own consistency, $\mathsf{H}_{<\omega}$ appears to be sufficiently natural. The theory $\mathsf{H}_{<\omega}$ is infinitely axiomatizable and proves existence of all individual hereditarily finite sets, but at the same time all its finite subtheories have finite models. Therefore, our example avoids the strong version of G\"odel second incompleteness theorem (due to Pudl\'ak) that asserts that no consistent a theory interpreting Robinson's arithmetic $\mathsf{Q}$ proves its own consistency. To show that $\mathsf{H}_{<\omega}$ proves its own consistency we establish a conservation result connecting Kalmar elementary arithmetic $\mathsf{EA}$ and $\mathsf{H}_{<\omega}$. We also consider the version of $\mathsf{H}_{<\omega}$ over higher order logic denoted $\mathsf{H}^{\omega}_{<\omega}$. It has the same non-G\"odelian property as $\mathsf{H}_{<\omega}$ but happens to be more attractive from a technical point of view. In particular, we show that $\mathsf{H}^{\omega}_{<\omega}$ proves a $\Pi_1$ sentence $\varphi$ of the predicate-only version of arithmetical language iff $\mathsf{EA}$ proves that $\varphi$ holds on the superexponential cut., Comment: 25 pages

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 quantifiers. This results motivates the general question of determining natural axioms concerning the truth predicate that can be added to $CT^{-}[PA]$ while maintaining conservativity over $PA$. Our main result shows that conservativity fails even for the extension of $CT^{-}[PA]$ obtained by the seemingly weak axiom of disjunctive correctness $DC$ that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, $CT^{-}[PA] + DC$ implies $Con(PA)$. Our main result states that the theory $CT^{-}[PA] + DC$ coincides with the theory $CT_0[PA]$ obtained by adding $\Delta_0$-induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cie\'sli\'nski (2010). For our proof we develop a new general form of Visser's theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (L\"ob's version of) G\"odel's second incompleteness theorem, rather than by using the Visser-Yablo paradox, as in Visser's original proof (1989)., Comment: 11 pages

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 $\Pi^1_1$ reflection strength order. We prove that there are no descending sequences of $\Pi^1_1$ sound extensions of $\mathsf{ACA}_0$ in this order. Accordingly, we can attach a rank in this order, which we call reflection rank, to any $\Pi^1_1$ sound extension of $\mathsf{ACA}_0$. We prove that for any $\Pi^1_1$ sound theory $T$ extending $\mathsf{ACA}_0^+$, the reflection rank of $T$ equals the proof-theoretic ordinal of $T$. We also prove that the proof-theoretic ordinal of $\alpha$ iterated $\Pi^1_1$ reflection is $\varepsilon_\alpha$. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.

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 $\mathbf T$ itself are bounded by a polynomial in $n$. At the same time he conjectures that $\mathbf T$ does not have polynomial proofs of the finite consistency statements $\operatorname{Con}(\mathbf T+\operatorname{Con}(\mathbf T))\!\restriction\!n$. In contrast we show that Peano arithmetic ($\mathbf{PA}$) has polynomial proofs of $\operatorname{Con}(\mathbf{PA}+\operatorname{Con}^*(\mathbf{PA}))\!\restriction\!n$, where $\operatorname{Con}^*(\mathbf{PA})$ is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also obtain a new proof of the result that the usual consistency statement $\operatorname{Con}(\mathbf{PA})$ is equivalent to $\varepsilon_0$ iterations of slow consistency. Our argument is proof-theoretic, while previous investigations of slow consistency relied on non-standard models of arithmetic.

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, there is a conservative extension $\beta$ of U in the expanded language, such that $\alpha\vdash\beta$ and $\beta\nvdash\alpha$. The result is preserved when we consider either extensions or model-conservative extensions of U in stead of conservative extensions. Moreover, the result is preserved when we replace $\vdash$ as ordering on the finitely axiomatized extensions in the expanded language by a special kind of interpretability, to wit interpretability that identically translates the symbols of the U-language. We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U., Comment: 15 pages

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 PSPACE-completeness., Comment: 7 pages

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 results we show that all linear orders that are interpretable in (N,+) are scattered orders with the finite Hausdorff rank and that the ranks are bounded in terms of the dimension of the respective interpretations. From our result about self-interpretations of PrA it follows that PrA isn't one-dimensionally interpretable in any of its finite subtheories. We note that the latter was conjectured by A. Visser., Comment: Published in proceedings of LFCS 2018

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 modal formula that made the formula unprovable PA if it were unprovable in GL. The arithmetical sentences for the evaluations were constructed using certain arithmetical fixed points. The method developed by Solovay have been used for establishing similar semantics for many other logics. In our proof we develop new more explicit construction of required evaluations that doesn't use any fixed points in their definitions. To our knowledge, it is the first alternative proof of the theorem that is essentially different from Solovay's proof in this key part., Comment: 13 pages, accepted to WoLLIC 2017 conference

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 $\mathsf{PA}+\mathrm{Con}_{\mathsf{s}}$ lies strictly between that of $\mathsf{PA}$ and $\mathsf{PA}$ together with its ordinary consistency: $\mathsf{PA}\subsetneq \mathsf{PA}+\mathrm{Con}_{\mathsf{s}}\subsetneq \mathsf{PA}+\mathrm{Con}$. This paper is a further investigation into slow provability and its interplay with ordinary provability in $\mathsf{PA}$. We study three variants of slow provability. The associated consistency statement of each of these yields a theory that lies strictly between $\mathsf{PA}$ and $\mathsf{PA}+\mathrm{Con}$ in terms of logical strength. We investigate Turing-Feferman progressions based on these variants of slow provability. We show that for our three notions, the Turing-Feferman progression reaches $\mathsf{PA}+\mathrm{Con}$ in a different numbers of steps, namely $\varepsilon_0$, $\omega$, and $2$. For each of the three slow provability predicates, we also determine its joint provability logic with ordinary $\mathsf{PA}$-provability., Comment: 46 pages

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 proof theory constructive systems of ordinal notations with fixed systems of cofinal sequences are used for the purposes of classification of provable recursive functions of theories. We show that any well-ordering from the hierarchy could be extended by a monadically definable system of cofinal sequences with Bachmann property. We show that the growth speed of functions from fast-growing hierarchy based on constructive ordinal notations from Caucal hierarchy may be only slightly influenced by the choice of monadically definable systems of cofinal sequences. We show that for ordinals less than $\omega^\omega$ a fast-growing hierarchy based on any system of ordinal notations from Caucal hierarchy coincides with L\"ob-Wainer hierarchy., Comment: 15 pages

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 free $\emptyset$-generated $GLP_{n}$ -algebras are decidable for all finite ordinals $n$., Comment: 39 pages

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 $\mathbf{GLP}$. They are closely related with the technique of ordinal analysis of $\mathbf{PA}$ and fragments of $\mathbf{PA}$ based on iterated reflection principles. We consider this notation system and it's fragments as structures with the signatures selected in a natural way. We prove that the full notation system and it's fragments, for ordinals ${\ge\omega_4}$, have undecidable elementary theories. We also prove that the fragments of the full system, for ordinals ${\le\omega_3}$, have decidable elementary theories. We obtain some results about decidability of elementary theory, for the ordinal notation systems with weaker signatures., Comment: 23 pages

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 , ,... . We show that, for every number n, the GLP-provability problem for formulas from L^n_0 is in PTIME., Comment: 12 pages, the results of this work and a proof sketch are in Advances in Modal Logic 2012 extended abstract on the same name