1,721,054 research outputs found
The Interpretability Logic of Peano Arithmetic
No full textWe extend Solovay's analysis of the provability logic of Peano Arithmetic, to the case of the interpretability logic
O-minimal spectra, infinitesimal subgroups and cohomology
No full textBy recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal expansion of an ordered field has a normal ``infinitesimal subgroup'' such that the quotient , equipped with the ``logic topology'', is a compact (real) Lie group. Our first result is that the functor sends exact sequences of definably compact groups into exacts sequences of Lie groups. We then study the connections between the Lie group and the o-minimal spectrum of . We prove that is a topological quotient of . We thus obtain a natural homomorphism from the cohomology of to the (\v{C}ech-)cohomology of . We show that if satisfies a suitable contractibility conjecture then is acyclic in \v{C}ech cohomology and is an isomorphism. Finally we prove the conjecture in some special cases
Sigma_n Interpretations of modal logic
No full textWe consider the predicative modal provability logic of Peano Arithmetic where the interpretation of each atomic modal formulas is required to be belong to the set of arithmetic formulas of complexity Sigma_0^n. We show that for distinct values of n the corresponding modal formulas form a strict hierarchy and each of them is complete for the class Pi^0_2
Factorization in generalized power series
No full textThe field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R((G≤0)) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): Sum_n t^(−1/n) + 1. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = (R, +, 0, ≤) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type α is either ω or of the form ω^ω^α and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case G = R. In the final part of the paper we study the irreducibility of series with finite support
Uniformly Approchable Functions and Spaces
No full textUniformly approachable (UA) functions are a common generalization of uniformly continuous functions an d perfect functions. We study UA-functions and UA-spaces i. e. those uniform spaces in which every real valued continuous function is UA. Such spaces properly include the UC-spaces (Atsuji spaces). We characterize the weakly-UA subspaces of the real line and give a new characterization of the UC spaces. We prove a topological result which implies, under the continuum hypothesis, the existence of a subset M of the the n-dimensional euclidean space R^n such that if two continuous functions f, g from R^n to R are are not constant on any open set and g(M) is a subset of f(M), then f=g
A note on coding techniques in bounded arithmetic
No full textTechnical Report n. 231, Dipartimento di Matematica, Via del Capitano 15, Sien
Combinatorial principles in elementary number theory
No full textWe prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give a new proof, which can be formalized in IΔ0 + Δ0EQ, of the fact that every prime of the form 4n + 1 is the sum of two squares
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