Arena Fernandez

In this paper we continue the study of the theories IΔ n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that IΔ n+1 (T) is Π n+2 –axiomatizable. In particular, IΔ n+1 (IΔ n+1 ) gives an axiomatization of Th Π n+2 (IΔ n+1 ) and is not finitely axiomatizable. This fact relates the fragment IΔ n+1 (IΔ n+1 ) to induction rule for Δ n+1 –formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons’ conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for Π n+2 and Δ n+1 formulas (see [2])

For a theory T, we study relationships among IΔ n +1 (T), LΔ n+1 (T) and B * Δ n+1 (T). These theories are obtained restricting the schemes of induction, minimization and (a version of) collection to Δ n+1 (T) formulas. We obtain conditions on T (T is an extension of B * Δ n+1 (T) or Δ n+1 (T) is closed (in T) under bounded quantification) under which IΔ n+1 (T) and LΔ n+1 (T) are equivalent. These conditions depend on Th Πn +2 (T), the Π n+2 –consequences of T. The first condition is connected with descriptions of Th Πn +2 (T) as IΣ n plus a class of nondecreasing total Π n –functions, and the second one is related with the equivalence between Δ n+1 (T)–formulas and bounded formulas (of a language extending the language of Arithmetic). This last property is closely tied to a general version of a well known theorem of R. Parikh. Using what we call Π n –envelopes we give uniform descriptions of the previous classes of nondecreasing total Π n –functions. Π n –envelopes are a generalization of envelopes (see [10]) and are closely related to indicators (see [12]). Finally, we study the hierarchy of theories IΔ n+1 (IΣ m ), m≥n, and prove a hierarchy theorem

We characterize the sets of all Π2 and all equation image theorems of IΠ−1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to equation image sentences cannot be extended to Πn + 2 sentences. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Truths will be defined as an agreement on uncertainties, the consensus over matters of empirical and social nature such as mathematics, physics or economics. As illustrated by Dennis Lindley , ‘individuals tend to know things to be true and false but the extent of this truth and falsity would always remain unknown’. Leading individuals to a permanent state of stress, uncertainty becomes a risk for the social community. Problems could not be presumed to be solvable as any kind of solution would be perceived as uncertain. The struggle for solutions would develop into a state of insecurity and conflict where no individual could trust the answers given by any other member of the community. To prevent a ‘war of all against all’ that would cause anxiety and permanent conflict, it will become an essential duty for the social community to diminish uncertainty by imposing truths. With those, individuals can be reassured about the certainty of their knowledge and use truths as a resolution tool for any dispute that might occur within the social space. For the purpose of social peace, truths will therefore be every outcome of the process through which uncertainty is transformed into an agreed form of knowledge. “Gravity”, “Death” and “God” will all be considered truths as they represent a form of agreement over different domains of the human and social life. It is not the intention of this essay to examine each of the truths separately but rather, to analyse them all together in order to formulate a hypothesis for all empirical and social matters.