Abstract
Suppose that \(\kappa \) is indestructibly supercompact and there is a measurable cardinal \(\lambda > \kappa \). It then follows that \(A_0 = \{\delta < \kappa \mid \delta \) is a measurable cardinal and the Mitchell ordering of normal measures over \(\delta \) is nonlinear\(\}\) is unbounded in \(\kappa \). If the Mitchell ordering of normal measures over \(\lambda \) is also linear, then by reflection (and without any use of indestructibility), \(A_1= \{\delta < \kappa \mid \delta \) is a measurable cardinal and the Mitchell ordering of normal measures over \(\delta \) is linear\(\}\) is unbounded in \(\kappa \) as well. The large cardinal hypothesis on \(\lambda \) is necessary. We demonstrate this by constructing via forcing two models in which \(\kappa \) is supercompact and \(\kappa \) exhibits an indestructibility property slightly weaker than full indestructibility but sufficient to infer that \(A_0\) is unbounded in \(\kappa \) if \(\lambda > \kappa \) is measurable. In one of these models, for every measurable cardinal \(\delta \), the Mitchell ordering of normal measures over \(\delta \) is linear. In the other of these models, for every measurable cardinal \(\delta \), the Mitchell ordering of normal measures over \(\delta \) is nonlinear.
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Notes
The terminology “lottery sum” is due to Hamkins, although the concept of the lottery sum of partial orderings has been around for quite some time and has been referred to at different junctures via the names “disjoint sum of partial orderings,” “side-by-side forcing,” and “choosing which partial ordering to force with generically.”
If \(\delta = \kappa \), then \({\dot{{{\mathbb {P}}}}}^\delta = {\dot{{{\mathbb {P}}}}}^\kappa \) is a term for trivial forcing.
Note that what we refer to as \({<} \gamma \)-strategically closed, [10] refers to as \(\gamma \)-strategically closed.
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The author’s research was partially supported by PSC-CUNY Grant 63505-00-51. The author wishes to thank the referee for helpful comments and suggestions which have been incorporated into the current version of the paper. In particular, the author is grateful to the referee for having pointed out a key clarification and correction to the proof of Lemma 2.2.
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Apter, A.W. Indestructibility and the linearity of the Mitchell ordering. Arch. Math. Logic 63, 473–482 (2024). https://doi.org/10.1007/s00153-024-00908-7
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DOI: https://doi.org/10.1007/s00153-024-00908-7
Keywords
- Supercompact cardinal
- Indestructibility
- Lottery sum
- Nonstationary support iteration
- Easton support iteration
- Mitchell ordering
- Ultrapower axiom (UA)