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Real-valued Measurable Cardinals and Well-orderings of the Reals

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Set Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

We show that the existence of atomlessly measurable cardinals is incompatible with the existence of well-orderings of the reals in L(ℝ), but consistent with the existence of well-orderings of the reals that are third-order definable in the language of arithmetic. Specifically, we provide a general argument that, starting from a measurable cardinal, produces a forcing extension where c is real-valued measurable and there is a Δ 22 -well-ordering of ℝ. A variation of this idea, due to Woodin, gives Σ 21 -well-orderings when applied to L[μ] or, more generally, Σ 21 (Hom∞) if applied to nice inner models, provided enough large cardinals exist in V. We announce a recent result of Woodin indicating how to transform this variation into a proof from large cardinals of the Ω-consistency of real-valued measurability of c together with the existence of Σ 21 -definable well-orderings of ℝ. It follows that if the Ω-conjecture is true, and large cardinals are granted, then this statement can always be forced.

However, we introduce a strengthening of real-valued measurability (real-valued hugeness), show its consistency, and prove that it contradicts the existence of third-order definable well-orderings of ℝ.

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Authors and Affiliations

  1. Department of Mathematics, California Institute of Technology, Mail code 253-37, Pasadena, CA, 91125, USA

    Andrés Eduardo Caicedo

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  1. Andrés Eduardo Caicedo
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Editors and Affiliations

  1. Departament de Lògica, Història I Filosofia de la Ciència, Universitat de Barcelona, Baldiri Reixac s/n, 08028, Barcelona, Spain

    Joan Bagaria

  2. Department of Mathematics, University of Toronto, Toronto, M5S 2E4, Canada

    Stevo Todorcevic

  3. C.N.R.S., UMR 7056, Université Paris 7, 2, Place Jussieu Case 7012, 75251, Paris Cedex 05, France

    Stevo Todorcevic

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Caicedo, A.E. (2006). Real-valued Measurable Cardinals and Well-orderings of the Reals. In: Bagaria, J., Todorcevic, S. (eds) Set Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7692-9_4

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