In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n , all translations of V -sentences are U -provably equivalent to sentences of complexity less than n . We call a sequential sentence with consistency power over T a pro-consistency statement for T (...) . We study pro-consistency statements. We provide an example of a pro-consistency statement for a sequential sentence A that is weaker than an ordinary consistency statement for A . We show that, if A is $${{\sf S}^{1}_{2}}$$ , this sentence has some further appealing properties, specifically that it is an Orey sentence for EA . The basic ideas of the paper essentially involve sequential theories. We have a brief look at the wider environment of the results, to wit the case of theories with pairing. (shrink)

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 has (...) decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles. (shrink)

We prove that a finitely generated group is context-free whenever its Cayley-graph has a decidable monadic second-order theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of context-free groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is context-free whenever its monadic second-order theory is decidable. Further, it is shown that the word (...) problem of a finitely generated group is decidable if and only if the first-order theory of its Cayley-graph is decidable. (shrink)

By PFM, we mean a finitely generated module over a principal ideal domain; a linear sentence is a sentence that contains no disjunctive and negative symbols. In this paper, we present an algorithm which decides the truth for linear sentences on a given PFM, and we discuss its time complexity. In particular, when the principal ideal domain is the ring of integers or a univariate polynomial ring over the field of rationals, the algorithm is polynomial-time. Finally, we consider some applications (...) to Abelian groups. (shrink)

We show in this paper that the theory of ordinal addition of any fixed ordinal ωα, with α less than ωω, admits a quantifier elimination. This in particular gives a new proof for the decidability result first established in 1965 by R. Büchi using transfinite automata. Our proof is based on the Ehrenfeucht games, and we show that quantifier elimination go through generalized power.RésuméOn montre ici que, pour tout ordinal α inférieur à ωω, la théorie additive de ωα admet une (...) élimination des quantificateurs. On obtient ainsi une nouvelle preuve de la décidabilité de ces théories . On utilise ici les jeux d'Ehrenfeucht, avec un formalisme à la Ferrante et Rackoff, et on montre que le fait d'admettre une élimination des quantificateurs se transmet par produit généralisé. (shrink)

The first-order logical theory Th $({\mathbb{N}},x + 1,F(x))$ is proved to be complete for the class ATIME-ALT $(2^{O(n)},O(n))$ when $F(x) = 2^{x}$ , and the same result holds for $F(x) = c^{x}, x^{c} (c \in {\mathbb{N}}, c \ge 2)$ , and F(x) = tower of x powers of two. The difficult part is the upper bound, which is obtained by using a bounded Ehrenfeucht–Fraïssé game.

A new method for obtaining lower bounds on the computational complexity of logical theories is presented. It extends widely used techniques for proving the undecidability of theories by interpreting models of a theory already known to be undecidable. New inseparability results related to the well known inseparability result of Trakhtenbrot and Vaught are the foundation of the method. Their use yields hereditary lower bounds . By means of interpretations lower bounds can be transferred from one theory to another. Complicated machine (...) codings are replaced by much simpler definability considerations, viz., the kinds of binary relations definable with short formulas on large finite sets. Numerous examples are given, including new proofs of essentially all previously known lower bounds for theories, and lower bounds for various theories of finite trees, which turn out to be particularly useful. (shrink)

We present familiar principles involving objects and classes (of objects), pairing (on objects), choice (selecting elements from classes), positive classes (elements of an ultrafilter), and definable classes (definable using the preceding notions). We also postulate the existence of a divine object in the formalized sense of lying in every definable positive class. ZFC (even extended with certain hypotheses just shy of the existence of a measurable cardinal) is interpretable in the resulting system. This establishes the consistency of mathematics relative to (...) the consistency of these systems. Measurable cardinals are used to interpret and prove the consistency of the system. Positive classes and various kinds of divine objects have played significant roles in theology. (shrink)

This volume announces a new era in the philosophy of God. Many of its contributions work to create stronger links between the philosophy of God, on the one hand, and mathematics or metamathematics, on the other hand. It is about not only the possibilities of applying mathematics or metamathematics to questions about God, but also the reverse question: Does the philosophy of God have anything to offer mathematics or metamathematics? The remaining contributions tackle stereotypes in the philosophy of religion. The (...) volume includes 35 contributions. It is divided into nine parts: 1. Who Created the Concept of God; 2. Omniscience, Omnipotence, Timelessness and Spacelessness of God; 3. God and Perfect Goodness, Perfect Beauty, Perfect Freedom; 4. God, Fundamentality and Creation of All Else; 5. Simplicity and Ineffability of God; 6. God, Necessity and Abstract Objects; 7. God, Infinity, and Pascal's Wager; 8. God and (Meta-)Mathematics; and 9. God and Mind. (shrink)