Correo Electrónico
DNINFOA - SIA
Bibliotecas
Convocatorias
Identidad U.N.
Publicado
Origen y desarrollo de la teoría de modelos núcleo I
Origin and development of the theory of core models I
DOI:
https://doi.org/10.15446/bol.mat.v31n1.121296Palabras clave:
teoría de conjuntos, modelos internos, teoría de modelos núcleo, grandes cardinales, sucesiones de medidas, teoremas de cubierta, extensores, sostenidos (es)Set theory, Inner Models, Core model theory, Large Cardinals, Sequences of measures, Covering Theorem, Extenders, Sharps (en)
Descargas
Se presenta una descripción del origen y desarrollo de la teoría de modelos núcleo hasta el modelo de Mitchell con sucesiones de medidas. Se describe la problemática asociada a ellos, técnicas, resultados y construcciones relacionados, así como las propiedades de grandes cardinales que los distinguen. Se discuten las fuentes apropiadas para seguir el desarrollo de la teoría, así como diversas referencias para sus aplicaciones. Al final se introducen los extensores para preparar (segunda parte) el desarrollo de modelos núcleo superiores.
We examine the origin and development of core model theory, up to Mitchell’s model of sequences of measures. We describe the difficulties associated with such models and related techniques, and constructions. Moreover, we address the large cardinals and covering theorems associated with these models. We discuss the appropriate bibliography to follow along with the development of the theory and its various applications. Towards the end, we introduce extenders to prepare the development of higher core models.
Referencias
[1] K. Devlin, Aspects of constuctibility, Springer-Verlag, Berl´ın, 1973.
[2] K. Devlin, Constructibility, Springer-Verlag, Berl´ın, 1984.
[3] K. Devlin and R. Jensen, Marginalia to a theorem of silver, En Logic Conference Kiel 1974 Springer-Verlag, Lecture Notes in Mathematics 499 (1975), 115-142.
[4] A. Dodd, The core model, Cambridge University Press, 1982.
[5] A. Dodd, Core models, J. Symbolic Logic 48 (1983), 78-90.
[6] A. Dodd and R. Jensen, The core model, Ann. Mathematical Log. 20 (1981), 43-75.
[7] A. Dodd and R. Jensen, The covering lemma for K, Ann. Mathematical Log. 22 (1982), 1-30.
[8] A. Dodd and R. Jensen, The covering lemma for L[U], Ann. Mathematical Log. 22 (1982), 127-135.
[9] H. D. Donder, Einige kombinatorische Ergebnisse in L, Ph.D. thesis, Inagural-Dissertation zur Erlangung der Doktorgrades, K¨oln, 1977.
[10] H. D. Donder, Regularity of ultrafilters and the core model, Israel J. Math. 63 (1988), 289-322.
[11] H. D. Donder, R. Jensen, and B. Koppelberg, Some applications of the core model, En Set theory and model theory (Bonn, 1979) of Lecture Notes in Math 872 (1981), 55-97.
[12] H. D. Donder, R. Jensen, and L. Stanley, Condensation-coherent global square systems, Procced. Symp. Pure Math. 42 (1985), 237-258.
[13] H. D. Donder and P. Koepke, On the consistency strength of "accessible” jonsson cardinals and of the weak chag conjecture, Ann. Pure App. Logic 25 (1983), 233-261.
[14] H. D. Donder, P. Koepke, and J. Levinski, Some stationary subsets of Pκ(λ), Proceed. Am. Math. Soc. 102 (1988), 1000-1004.
[15] H. D. Donder and J. Levinski, On weakly precipitous filters, Israel J. Math. 67 (1989), 225-242.
[16] H. D. Donder and J. Levinski, Some principles related to Chang’s conjecture, Ann. Pure App. Logic 45 (1989), 39-101.
[17] H. Gaifman, Elementary embeddings of models of set-theory and certain subtheories, En Axiomatic set theory (Proc. Sympos. Pure Math. Vol. XIII, Part II, Univ. 24 California, Los Angeles, Calif. 1967) Amer. Math. Soc. Providence R.I. (1974), 33-101.
[18] A. Gallardo Quiroz, E. A. Valenzuela Nuncio, and L. M. Villegas Silva, Modelo núcleo para medidas de orden cero, En preparación.
[19] A. Gallardo Quiroz, E. A. Valenzuela Nuncio, and L. M. Villegas Silva, Modelos internos para cardinales medibles, sucesiones de medidas y cardinales fuertes, Propuesto para publicación.
[20] K. Gödel, The consistency of the axiom of choice and of the generalized continuum hypothesis, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 556-557.
[21] K. Gödel, Consistency-proof for the generalized continuum-hypothesis, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 220-224.
[22] K. Gödel, The consistency of the continuum hypothesis, Annals of Mathematics Studies 3 (1940).
[23] A. Hajnal, On a consistency theorem connected with the generalized continuum problem, Z. Math. Logik Grundlagen Math. 2 (1956), 131-136.
[24] A. Hajnal, On a consistency theorem connected with the generalized continuum problem, Acta Math. Acad. Sci. Hungar 12 (1961), 321-376.
[25] R. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229-308.
[26] R. Jensen, Manuscript on fine structure, inner model theory, and the core model below one Woodin cardinal, Book in preparation, https://ivv5hpp.uni-muenster.de/u/rds/jensenskript march 19 2024.pdf.
[27] R. Jensen, Measures of order zero, Diversos PDF en https://www.mathematik.hu-berlin.de/∼raesch/org/jensen.html.
[28] R. Jensen, Some remarks on below 0, manuscrito no publicado.
[29] R. Jensen and M. Zeman, Smooth categories and global , Ann. Pure Appl. Logic 102 (2000), 101-138.
[30] A. Kanamori, The higher infinite, Springer-Verlag, 2nd Ed., 2009.
[31] P. Koepke and P. Welch, Global square and mutual stationarity at the ℵn, Ann. Pure Appl. Logic 162 (2011), 787-806.
[32] K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179-227.
[33] A. Levy, Ind´ependance conditionnelle de V = L et d’axiomes qui se rattachent au syst`eme de M. G¨odel, C. R. Acad. Sci. Paris 245 (1957), 1582-1583.
[34] A. Levy, A generalization of G¨odel’s notion of constructibility, J. Symbolic Logic 25 (1960), 147-155.
[35] A. R. D. Mathias, Weak systems of Gandy, Jensen and Devlin, En Set Theory: Centre de Recerca Matematica, Barcelona 2003-4 edited by Joan Bagaria and Stevo Todorˇcevi´c, Trends in Mathematics, Birkh¨auser Verlag, Basel, 25 (2006), 149-224.
[36] W. Mitchell, Sets constructible from sequences of ultrafilters, J. Symbolic Logic 39 (1974), 57-66.
[37] W. Mitchell, Hypermeasurable cardinals, Logic Coll. ’78, North-Holland, Amsterdam (1979), 303-317.
[38] W. Mitchell, Sets constructed from sequences of measures: revisited, J. Symbolic Logic 48 (1983).
[39] W. Mitchell, The core model for sequences of measures I, Math. Proceed. Cambridge Phil Soc. 95 (1984), 229-260.
[40] W. Mitchell, Handbook of set theory, ch. 18. The covering lemma, pp. 1497-11594, M. Foreman, A. Kanamori, editors, Springer-Verlag, 2010.
[41] J. A. Nido Valencia, H. G. Salazar Pedroza, and L. M. Villegas Silva, Modelos internos, grandes cardinales y álgebra, Universidad Autónoma de la Ciudad de México, Propuesto para publicación.
[42] R. Schindler, Set theory, exploring independence and truth, Springer-Verlag, Switzerland, 2014.
[43] R. Schindler and M. Zeman, Handbook of set theory, ch. 9. Fine Structure, pp. 605-656, M. Foreman, A. Kanamori, editors, Springer-Verlag, 2010.
[44] D. Scott, Measurable cardinals and constructible sets, Bulletin de l’Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques 9 (1961), 521-524.
[45] J. Silver, Some applications of model theory in set theory, Ph.D. thesis, Univ. California, Berkeley, 1966.
[46] J. Silver, A large cardinal in the constructible universe, Fundamenta Mathematicae 69 (1970), no. 1, 93-100.
[47] R. M. Solovay, Strongly compact cardinals and the GCH, Proceedings of the 1971 Tarski Symposium 25 (1974), 365-372.
[48] L. Stanley, Surveys in set theory, ch. 7. A short course on Gap-one morasses with a review of the Fine structure of L, pp. 197-243, Cambridge University Press, 1983.
[49] P. Welch, Combinatorial principles in the core model, Ph.D. thesis, Oxford University, 1983.
[50] P. Welch, Handbook of set theory, ch. 10. Σ∗-Fine Structure, pp. 657-736, M. Foreman, A. Kanamori, editors, Springer-Verlag, 2010.
[51] M. Zeman, Inner models and large cardinals, Walter de Gruytier, 2002.
