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# INNER MODEL THEORETIC GEOLOGY

Abstract

One of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P-constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ω (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model.

References
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[1] David, R., A very absolute ${\rm{\Pi }}_2^1$ real singleton . Annals of Mathematical Logic, vol. 23 (1982), pp. 101120.
[2] Fuchs, G., Closed maximality principles: implications, separations and combinations, this Journal, vol. 73 (2008), no. 1, pp. 276308.
[3] Fuchs, G., Hamkins, J. D., and Reitz, J., Set-theoretic geology . Annals of Pure and Applied Logic, vol. 166 (2015), no. 4, pp. 464501.
[4] Fuchs, G. and Schindler, R., The solidity and nonsolidity of initial segments of the core model, in preparation.
[5] Hamkins, J. D., A simple maximality principle, this Journal, vol. 68 (2003), no. 2, pp. 527550.
[6] Hamkins, J. D., Kirmayer, G., and Perlmutter, N. L.. Generalizations of the Kunen Inconsistency . Annals of Pure and Applied Logic, vol. 163 (2012), no. 12, pp. 18721890.
[7] Laver, R., Certain very large cardinals are not created in small forcing extensions. Annals of Pure and Applied Logic, vol. 149 (2007), pp. 16.
[8] Mitchell, W. J. and Steel, J. R., Fine Structure and Iteration Trees, Lecture Notes in Logic, vol. 3, Springer, 1994.
[9] Mitchell, W. and Schindler, R., A universal extender model without large cardinals in V, this Journal, vol. 69 (2004), no. 2, pp. 371386.
[10] Schlutzenberg, F., Measures in mice, Ph.D. thesis, University of Berkeley, Berkeley, CA, 2007.
[11] Schindler, R. and Steel, J., The self-iterability of L[E], this Journal, vol. 74 (2009), no. 3, pp. 751779.
[12] Schindler, R., Uhlenbrock, S., and Woodin, W. H.. Mice with finitely many Woodin cardinals from optimal determinacy hypotheses, in preparation.
[13] Steel, J. R., The Core Model Iterability Problem, Lecture Notes in Logic, vol. 8, Springer, Berlin, 1996.
[14] Steel, J. R., An outline of inner model theory , Handbook of Set Theory (Foreman, M., Kanamori, A., and Magidor, M., editors), Springer, Berlin, 2009.
[15] Steel, J. R. and Woodin, W. H.. HOD as a core model, to appear.
[16] Woodin, W. H., Davis, J., and Rodriguez, D., The HOD dichotomy. Notes of the Apalachian Set Theory meeting 2012 at Cornell, unpublished, pages 1–19, 2012, available at http://www.math.cmu.edu/∼eschimme/Appalachian/WoodinDavisRodriguez.pdf .
[17] Hugh Woodin, W., The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture, Proceedings of the Conference on the Continuum in Philosophy and Mathematics , 2004.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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