We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results (...) are generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman. (shrink)

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+1-formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first-order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, (...) we obtain results on the quantifier complexity, finite axiomatizability and relative strength of schemes for Δn+1-formulas. (shrink)

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying (...) second-order logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. (shrink)

There has been a recent interest in hierarchical generalizations of classic incompleteness results. This paper provides evidence that such generalizations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “$\mathrm {T}$is$\Sigma _n$-ill” (...) is a canonical example of a$\Sigma _{n+1}$formula that is$\Pi _{n+1}$-conservative over$\mathrm {T}$. (shrink)

We define classes Φnof formulae of first-order arithmetic with the following properties:(i) Everyφϵ Φnis classically equivalent to a Πn-formula (n≠ 1, Φ1:= Σ1).(ii)(iii)IΠnandiΦn(i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φnboth under existential and universal quantification (we call these classes Θn) the corresponding theoriesiΘnstill prove the same Π2-formulae. In a second part we (...) consideriΔ0plus collection-principles. We show that both the provably recursive functions and the provably total functions ofare polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence. (shrink)

Global properties of canonical derivability predicates in Peano Arithmetic) are studied here by means of a suitable propositional modal logic GL. A whole book [1] has appeared on GL and we refer to it for more information and a bibliography on GL. Here we propose a sequent calculus for GL and, by exhibiting a good proof procedure, prove that such calculus admits the elimination of cuts. Most of standard results on GL are then easy consequences: completeness, decidability, finite model (...) property, interpolation and the fixed point theorem. (shrink)

We carry out a study of definability issues in the standard models of Presburger and Skolem arithmetics (henceforth referred to simply as Presburger and Skolem arithmetics, for short, because we only deal with these models, not the theories, thus there is no risk of confusion) supplied with free unary predicates—which are strongly related to definability in the monadic SOA (second-order arithmetic) without × or + , respectively. As a consequence, we obtain a very direct proof for ${\Pi^1_1}$ -completeness of (...) Presburger, and also Skolem, arithmetic with a free unary predicate, generalize it to all ${\Pi^1_n}$ -levels, and give an alternative description of the analytical hierarchy without × or + . Here ‘direct’ means that one explicitly m-reduces the truth of ${\Pi^1_1}$ -formulae in SOA to the truth in the extended structures. Notice that for the case of Presburger arithmetic, the ${\Pi^1_1}$ -completeness was already known, but the proof was indirect and exploited some special ${\Pi^1_1}$ -completeness results on so-called recurrent nondeterministic Turing machines—for these reasons, it was hardly able to shed any light on definability issues or possible generalizations. (shrink)

We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, and for any $n\geq k+1$, $$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}} \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}, \end{align*}$$ $$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}} \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}^{-}, \end{align*}$$ where T is $T_0$ augmented with full induction, and $\mathrm {TI}_{\varPi ^1_n}^{-}$ denotes the schema of (...) transfinite induction up to $\varepsilon _0$ for $\varPi ^1_n$ formulas without set parameters. (shrink)

We study the relative strength of the two axioms Every Pell equation has a nontrivial solution Exponentiation is total over weak fragments, and we show they are equivalent over IE1. We then define the graph of the exponential function using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol #, where # = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment.

This paper presents proof that Buss's S22 can prove the consistency of a fragment of Cook and Urquhart's PV from which induction has been removed but substitution has been retained. This result improves Beckmann's result, which proves the consistency of such a system without substitution in bounded arithmetic S12. Our proof relies on the notion of "computation" of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and (...) either its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven. This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss's S12. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies. (shrink)

We consider the problem whether the $\forall \Sigma _{1}^{b}$ theorems of the fragments $T_{2}^{a}$ form a strictly increasing hierarchy. We shall show a link to some results about the lengths of proofs in predicate logic that supports the conjecture that the hierarchy is strictly increasing.

Let T 0?T 1 denote that each computable function, which is provable total in the first order theory T 0, is also provable total in the first order theory T 1. Te relation ? induces a degree structure on the sound finite Π2 extensions of EA (Elementary Arithmetic). This paper is devoted to the study of this structure. However we do not study the structure directly. Rather we define an isomorphic subrecursive degree structure <≤,?>, and then we study <≤,?> (...) by ubrecursive and computability-theoretic means. Furthermore, we introduce and investigate some operators on the degrees of <≤,?>. These operators corresponds to inferencerules in formal arithmetic. One operator corresponds to the Σ1 collection rule. Another operator corresponds to the Σ1 induction rule. (shrink)

We give combinatorial and computational characterizations of the NP search problems definable in the bounded arithmetic theories $T_{2}^{2}$ and $T_{3}^{2}$.

Solovay proved (Israel J Math 25(3–4):287–304, 1976) that the propositional provability logic of any ∑2-sound recursively enumerable extension of PA is characterized by the propositional modal logic GL. By contrast, Montagna proved in (Notre Dame J Form Log 25(2):179–189, 1984) that predicate provability logics of Peano arithmetic and Bernays–Gödel set theory are different. Moreover, Artemov proved in (Doklady Akademii Nauk SSSR 290(6):1289–1292, 1986) that the predicate provability logic of a theory essentially depends on the choice of a binumeration of (...) the theory which is used to construct the provability predicate. In this paper, we compare predicate provability logics of I∑ n ’s. For a binumeration α(x) of a recursive theory T, let PL α(T) be the predicate provability logic of T defined by α(x). We prove that for any natural numbers i, j such that 0 < i < j, there exists a ∑1 binumeration α(x) of some recursive axiomatization of I∑ i such that ${{\sf PL}_\alpha({\rm I \Sigma}_i) \nsupseteq \bigcap_{\beta(x)}{\sf PL}_\beta({\rm I \Sigma}_j)}$ and ${{\sf PL}_\alpha({\rm I \Sigma}_i) \nsubseteq \bigcup_{\beta(x)}{\sf PL}_\beta({\rm I \Sigma}_j)}$ , where β(x) ranges over all ∑1 binumerations of recursive axiomatizations of I∑ j. (shrink)

We study the minimal enumeration degree problem in models of fragments of Peano arithmetic () and prove the following results: in any model M of Σ2 induction, there is a minimal enumeration degree if and only if M is a nonstandard model. Furthermore, any cut in such a model has minimal e-degree. By contrast, this phenomenon fails in the absence of Σ2 induction. In fact, whether every Σ2 cut has minimal e-degree is independent of the Σ2 bounding principle.

In this paper we give a new proof of the characterization of the closed fragment of the provability logic of Heyting's Arithmetic. We also provide a characterization of the closed fragment of the provability logic of Heyting's Arithmetic plus Markov's Principle and Heyting's Arithmetic plus Primitive Recursive Markov's Principle.

We study the long-standing open problem of giving$\forall {\rm{\Sigma }}_1^b$separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the$\forall {\rm{\Sigma }}_1^b$Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes (...) the surjective weak pigeonhole principle for FPNPfunctions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of$T_2^1$augmented with the surjective weak pigeonhole principle for polynomial time functions. (shrink)

The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that (...) the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations. (shrink)

Samuel R. Buss and Aleksandar Ignjatović. Unprovability of Consistency Statements in Fragments of Bounded Arithmetic.

Modifying the methods of Z. Adamowicz's paper Herbrand consistency and bounded arithmetic [3] we show that there exists a number n such that ⋃m Sm (the union of the bounded arithmetic theories Sm) does not prove the Herbrand consistency of the finitely axiomatizable theory $S_{3}^{n}$.

In this paper we consider some fragments of $$\textsf{IOpen}$$ (Robinson arithmetic $$\mathsf Q$$ with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that $$\mathsf {I(lit)}$$ is equivalent to $$\textsf{IOpen}$$ and is not finitely axiomatizable over $$\mathsf Q$$, establish some inclusion relations between $$\mathsf {I(=)}, \mathsf {I(\ne )}, \mathsf {I(\leqslant )}$$ and $$\textsf{I} (\nleqslant )$$. We also prove that the set of diophantine equations solvable in models of $$\mathsf (...) I (=)$$ is (algorithmically) decidable. (shrink)

A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary (...) formulae increases the strength by one inductive definition. (shrink)

The philosophy of arithmetic of Wittgenstein's Tractatus is outlined and the central role played in it by the general notion of operation is pointed out. Following which, the language, the axioms and the rules of a formal theory of operations, extracted from the Tractatus, are presented and a theorem of interpretability of the equational fragment of Peano's Arithmetic into such a formal theory is proven.

The complexity class of [Formula: see text]-polynomial local search problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the [Formula: see text]-definable functions of [Formula: see text] are characterized in terms of [Formula: see text]-PLS problems. These [Formula: see text]-PLS problems can be defined in a weak base theory such as [Formula: see text], and proved to be total in [Formula: see text]. Furthermore, (...) the [Formula: see text]-PLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new [Formula: see text]-principle that is conjectured to separate [Formula: see text] and [Formula: see text]. (shrink)

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 Π1-sentence such that IΔ0 cannot prove its Herbrand Consistency.

We show that for each n ≥ 1, if T2n does not prove the weak pigeonhole principle for Σbn functions, then the collection scheme B Σ1 is not finitely axiomatizable over T2n. The same result holds with Sn2 in place of T 2n.

We characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ 0 + EXP, PRA); (PRA, IΣ 1 ); (IΣ m , IΣ n ) for $1 \leq m etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.

We prove that the standard cut is definable in each existentially closed model ofIΔ0+ exp by a (parameter free) П1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic.

We introduce a nonstandard arithmetic $NQA^-$ based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by $NQA^+$ ), with a weakened external open minimization schema. A finitary consistency proof for $NQA^-$ formalizable in PRA is presented. We also show interesting facts about the strength of the theories $NQA^-$ and $NQA^+$ ; $NQA^-$ is mutually interpretable with $I\Delta_0 + EXP$ , and on the other hand, $NQA^+$ interprets the theories IΣ1 and $WKL_0$.

For any n, we construct a model of in which each formula is equivalent to an formula.

In 1893, Schatunowsky showed that 30 is the largest number all of whose totatives are primes; we show that this result cannot be proved, in any form, in Chebyshev's theorem (Bertrand's postulate), even if all irreducibles are primes. Bonse's inequality is shown to be indeed weaker than Chebyshev's theorem. Schatunowsky's theorem holds in together with Bonse's inequality, the existence of the greatest prime dividing certain types of numbers, and the condition that all irreducibles be prime.

In the first part of this paper we investigate the intuitionistic version $iI\!\Sigma_1$ of $I\!\Sigma_1$ (in the language of $PRA$ ), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for $HA$ and establishes a number of nice properties for $iI\!\Sigma_1$ , e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically (...) than classically: $iI\!\Sigma_1$ together with induction over arbitrary prenex formulas is $\Pi_2$ -conservative over $iI\!\Pi_2$ . In the second part of the article we study the relation of $iI\!\Sigma_1$ to $iI\!\Pi_1$ (in the usual arithmetical language). The situation here is markedly different from the classical case in that $iI\!\Pi_1$ and $iI\!\Sigma_1$ are mutually incomparable, while $iI\!\Sigma_1$ is significantly stronger than $iI\!\Pi_1$ as far as provably recursive functions are concerned: All primitive recursive functions can be proved total in $iI\!\Sigma_1$ whereas the provably recursive functions of $iI\!\Pi_1$ are all majorized by polynomials over ${\Bbb N}$ . 0 $iI\!\Pi_1$ is unusual also in that it lacks closure under Markov's Rule $\mbox{MR}_{PR}$. (shrink)

By algebraic means, we give an equational axiomatization of the equational fragments of various systems of arithmetic. We also introduce a faithful semantics according to which, for every reasonable system T for arithmetic, there is a model where exactly the theorems of T are true.

In this paper I give a characterization of the closed fragment of the provability logic of $I \triangle_0 + \mathrm{EXP}$ with a propositional constant for $\mathrm{EXP}$. In three appendices many details on arithmetization are provided.

In this paper we carry out a comparative study of $\mathrm{I}\Sigma_1$ and PRA. We will in a sense fully determine what these theories have to say about each other in terms of provability and interpretability. Our study will result in two arithmetically complete modal logics with simple universal models.

Recently, Feferman and Hellman (and Aczel) showed how to establish the existence and categoricity of a natural number system by predicative means given the primitive notion of a finite set of individuals and given also a suitable pairing function operating on individuals. This short paper shows that this existence and categoricity result does not rely (even indirectly) on finite-set induction, thereby sustaining Feferman and Hellman's point in favor of the view that natural number induction can be derived from a very (...) weak fragment of finite-set theory, so weak that finite-set induction is not assumed. Many basic features of finiteness fail to hold in these weak fragments, conspicuously the principle that finite sets are in one-one correspondence with a proper initial segments of a (any) natural number structure. In the last part of the paper, we propose two prima facie evident principles for finite sets that, when added to these fragments, entail this principle. (shrink)