Abstract
In this paper we are concerned about the ways GCH can fail in relation to rank-into-rank hypotheses, i.e., very large cardinals usually denoted by I3, I2, I1 and I0. The main results are a satisfactory analysis of the way the power function can vary on regular cardinals in the presence of rank-into-rank hypotheses and the consistency under I0 of the existence of \({j : V_{\lambda+1} {\prec} V_{\lambda+1}}\) with the failure of GCH at \({\lambda}\).
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References
Corazza P.: The wholeness axiom and Laver sequences. Ann. Pure Appl. Logic 105, 157–260 (2000)
Corazza P.: Lifting elementary embeddings \({j : V_\lambda \to V_\lambda}\). Arch. Math. Logic 46, 61–72 (2007)
Cummings J., Foreman M.: Diagonal prikry extensions. J. Symb. Logic 75, 1383–1402 (2010)
Easton W.B.: Powers of regular cardinals. Ann. Math. Logic 1, 139–178 (1970)
Friedman, S.-D.: Large cardinals and L-like universes. In: Andretta, A. (ed.) Set Theory: Recent Trends and Applications. Quaderni di Matematica, vol. 17, pp. 93–110 (2006)
Gitik M.: Blowing up the power of a singular cardinal—wider gaps. Ann. Pure Appl. Logic 116, 1–38 (2002)
Gitik, M.: Prikry-type forcings. In: Foreman, M. Kanamori, A. (eds.) Handbook of Set Theory, pp. 1351–1447. Springer, The Netherlands (2010)
Hamkins J.: Fragile measurability. J. Symb. Logic 59, 262–282 (1994)
Hausdorff F.: Grundzüge einer Theorie der geordneten Mengen. Math. Ann. 65, 435–505 (1908)
Kanamori A.: The Higher Infinite. Springer, Berlin (1994)
Kunen K.: Elementary embeddings and infinite combinatorics. J. Symb. Logic 36, 407–413 (1971)
Laver R.: Implications between strong large cardinal axioms. Ann. Pure Appl. Logic 90, 79–90 (1997)
Laver R.: Reflection of elementary embedding axioms on the \({L[V_{\lambda+1}]}\) hierarchy. Ann. Pure Appl. Logic 107, 227–238 (2001)
Mathias A.R.D.: On sequences generic in the sense of Prikry. J. Aust. Math. Soc. 15, 409–414 (1973)
Scott D.: Measurable cardinals and constructible sets. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9, 521–524 (1961)
Silver, J.: On the singular cardinals problem. In: Proceedings of the International Congress of Mathematicians, Vancouver, pp. 265–268 (1974)
Solovay, R.: Strongly compact cardinals and the GCH. In: Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics, vol. 25, pp. 365–372. American Mathematical Society, Providence (1974)
Woodin W.H.: Suitable extender models II: beyond ω-huge. J. Math. Logic 11, 115–436 (2011)
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Dimonte, V., Friedman, SD. Rank-into-rank hypotheses and the failure of GCH. Arch. Math. Logic 53, 351–366 (2014). https://doi.org/10.1007/s00153-014-0369-8
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DOI: https://doi.org/10.1007/s00153-014-0369-8