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A model of Peano Arithmetic is short recursively saturated if it realizes all its bounded finitely realized recursive types. Short recursively saturated models of are exactly the elementary initial segments of recursively saturated models of . In this paper, we survey and prove results on short recursively saturated models of and their automorphisms. In particular, we investigate a certain subgroup of the automorphism group of such models. This subgroup, denoted , contains all the automorphisms of a countable short recursively saturated model of which can be extended to an automorphism of the countable recursively saturated elementary end extension of the model.
Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert's style of proof and Gentzen's deductive systems. In this context we call it CED (Calculus of Equational Deduction). This deductive method promotes logical equivalence over implication and shows that there are easy ways to prove predicate formulas without the introduction of hypotheses or metamathematical tools such as the deduction theorem. Moreover, syntactic considerations (in Dijkstra's words, "letting the symbols do the work") have led to the "calculational style," an impressive array of techniques for elegant proof constructions. In this paper, we formalize intuitionistic predicate logic according to CED with similar success. In this system (I-CED), we prove Leibniz's principle for intuitionistic logic and also prove that any (intuitionistic) valid formula of predicate logic can be proved in I-CED.
We present and show topological completeness for LB, the logic of the topological border. LB is also a logic of epistemic ignorance. Also, we present and show completeness for LUT, the logic of unknown truths. A simple topological completeness proof for S4 is also presented using a T1 space
We consider a problem of constructing a model that omits given complete types. We present two results. The first one is related to the Lopez-Escobar theorem and the second one is a version of Morley's omitting types theorem.
In this paper we propose substructural propositional logic obtained by da Costa weakening of the intuitionistic negation. We show that the positive fragment of the da Costa system is distributive lattice logic, and we apply a kind of da Costa weakening of negation, by preserving, differently from da Costa, its fundamental properties: antitonicity, inversion, and additivity for distributive lattices. The other stronger paraconsistent logic with constructive negation is obtained by adding an axiom for multiplicative property of weak negation. After that, we define Kripke-style semantics based on possible worlds and derive from it many-valued semantics based on truth-functional valuations for these two paraconsistent logics. Finally, we demonstrate that this model-theoretic inference system is adequate—sound and complete with respect to the axiomatic da Costa-like systems for these two logics.
Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums (with , , ) and (with positive integers) are never integers, are shown to hold in , a very weak arithmetic, whose axiom system has no induction axiom.
A famous theorem by van der Waerden states the following: Given any finite coloring of the integers, one color contains arbitrarily long arithmetic progressions. Equivalently, for every q,k, there is an N = N(q,k) such that for every q-coloring of an interval of length N one color contains a progression of length k. An obvious question is what is the growth rate of N = N(q,k). Some proofs, like van der Waerden's combinatorial argument, answer this question directly, while the topological proof by Furstenberg and Weiss does not. We present an analysis of (Girard's variant of) Furstenberg and Weiss's proof based on monotone functional interpretation, both yielding bounds and providing a general illustration of proof mining in topological dynamics. The bounds do not improve previous results by Girard, but only—as is also revealed by the analysis—because the combinatorial proof and the topological dynamics proof in principle are identical.