Local induction and provably total computable functions

Let I¦− 2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free ¦2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions of I¦− 2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of certain local variants of induction principles closely related to I¦− 2 . In this way, we obtain a more direct answer to Kaye’s question, avoiding the metamathematical machinery (reflection principles, provability logic,...) needed for Beklemishev’s original proof. Our methods are model–theoretic and allow for a general study of I¦− n+1 for all n ¸ 0. In particular, we derive a new conservation result for these theories, namely that I¦− n+1 is ¦n+2–conservative over I§n for each n ¸ 1.Ministerio de Ciencia e Innovación MTM2008–06435Ministerio de Ciencia e Innovación MTM2011–2684

Local Induction and Provably Total Computable Functions: A Case Study

Let IΠ−2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free Π2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions (p.t.c.f.) of IΠ−2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of the p.t.c.f. of certain local versions of induction principles closely related to IΠ−2 . This analysis is essentially based on the equivalence between local induction rules and restricted forms of iteration. In this way, we obtain a more direct answer to Kaye’s question, avoiding the metamathematical machinery (reflection principles, provability logic,...) needed for Beklemishev’s original proof.Ministerio de Ciencia e Innovación MTM2008–0643

Social security and retirement in Spain.

This paper examines the interplay between the retirement incentives generated by the Spanish public pension system and the decision to early retire from the labor force. For many workers, particularly those with below--median earning profiles or with incomplete working histories, retirement at ages earlier then 65 appears to be the most rational strategy. Indeed we calculate that, exception made for workers in the upper twenty percent of the earnings distribution, the current system generates strong incentives to retire before the age of 65. We identify the minimum pension rule, embedded in the Spanish legislation, as the most important source of such incentives. We apply the insights from our study to a preliminary evaluation of the reform legislation just enacted (July 26, 1997). We find it doubtful that it may seriously weaken such incentives. On the contrary, the final outcome may be an increase in the proportion of workers for which the minimum pension is binding, thereby increasing the incentive to early retirement.Retirement; Pensions; incentives;

Existentially Closed Models in the Framework of Arithmetic

We prove that the standard cut is definable in each existentially closed model of IΔ0 + 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.Ministerio de Educación y Ciencia MTM2011–2684

Potencial emotional problems of teacher

En mi opinión, el mundo de la educación vive en la actualidad circunstancias muy graves y muy difíciles de soportar con el equilibrio emocional necesario para integridad personal de los docentes y para el desarrollo de su función. Me parece que en ningún otro momento histórico las circunstancias educativas han sido tan graves. Entiendo que mi argumentación no puede rebasar el nivel de hipótesis más o menos fundamentadas. Los estudios de las épocas los suele hacer la Historia, cuando ya no pueden corregirse los errores y cuando sus gestores –los historiadores– pueden interpretar impunemente porque no se les pueden rebelar, no les pueden pedir cuentas los gestores de los hechos. Hablar del presente ha sido función de profetas y éstos concitan el rechazo social. Teóricamente sería función de sociólogos, pero no parece que a la Sociología le interese abordar las problemáticas a las que nos pretendemos referir. Parece que a la Sociología oficial sólo le motiva la confirmación de las hipótesis que le interesan. La educación se encuentra en la encrucijada del principio de realidad y del principio de placer a los que se refiere Sigmund Freud. El principio de placer se manifiesta en la hipótesis de que la educación más favorecedora de las personas con menos recursos es la que se puede obtener sin grandes esfuerzos por parte de ellas, a la que llamamos “igualitaria”. No parece aceptable por el discurso común decir que lo más progresista, lo que cambia la realidad de privilegios de los pocos en otra de oportunidades para los más desfavorecidos, es una educación exigente en la que éstos puedan demostrar y ejercer su valía. También hemos de negar que la orientación hacia la formación profesional a los 14 años, como hacen, de manera mejorable, los actuales programas de cualificación profesional inicial, pueda condicionar negativamente el futuro de unos alumnos, cuando, por el contrario, para éstos puede establecer una vía de desarrollo de la capacidad de abstracción, que les bloquea la mayor teorización de la educación general, abriéndoseles, pues, con la formación profesional un buen camino hacia las ingenierías. La educación, desde mi punto de vista vive, por una parte, en tensión; por otra, para evitarla, se convierte en teatro de simulaciones en el que se producen situaciones esquizoides que llevan al profesorado a posibles problemas emocionales

Interferencias lingüísticas en el aprendizaje de español por estudiantes francófonos

L'apprentissage de l'espagnol par des étudiants \ud francophones a mis en évidence une série d'interférences \ud linguistiques communes aux étudiants francophones, \ud indépendement de leur nationalité (française, belge, \ud canadienne, suisse...). \ud Au niveau lexico-sémantique, ils produisent certains \ud vocables, dans une langue artificielle que nous appellerons \ud FRANPAÑOL, pour faciliter la communication verbale entre eux, \ud mais qui n'existent pas dans la langue espagnole. Dans ce \ud travail nous ferons une étude diagnostique au niveau \ud grammatical et léxico-semantique

On Rules and Parameter Free Systems in Bounded Arithmetic

We present model–theoretic techniques to obtain conservation results for first order bounded arithmetic theories, based on a hierarchical version of the well known notion of an existentially closed model.Ministerio de Educación y Ciencia MTM2005-0865

Existentially Closed Models and Conservation Results in Bounded Arithmetic

We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories Si2 and Ti2 and prove that they are ∀Σbi conservative over their inference rule counterparts, and ∃∀Σbi conservative over their parameter-free versions. A similar analysis of the Σbi-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.Ministerio de Educación y Ciencia MTM2005-08658Junta de Andalucía TIC-13

Envelopes, indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that BΣn +1 (the fragment of Arithmetic given by the collection scheme restricted to Σn +1‐formulas) is a Πn +2‐conservative extension of IΣn (the fragment given by the induction scheme restricted to Σn ‐formulas). In this paper, as a continuation of our previous work on collection schemes for Δn +1(T )‐formulas (see [4]), we study a general version of this theorem and characterize theories T such that T + BΣn +1 is a Πn +2‐conservative extension of T . We prove that this conservativeness property is equivalent to a model‐theoretic property relating Πn ‐envelopes and Πn ‐indicators for T . The analysis of Σn +1‐collection we develop here is also applied to Σn +1‐induction using Parsons' conservativeness theorem instead of Friedman‐Paris' theorem. As a corollary, our work provides new model‐theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos (see [8]): BΣn +1 and IΣn +1 are Σn +3‐conservative extensions of their parameter free versions, BΣ–n +1 and IΣ–n +1.Junta de Andalucía TIC-13