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Strong Hypotheses

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Abstract

This chapter provides the basic theory for the strong hypotheses that generalize measurability. §22 discusses Solovay and Reinhardt’s concept of supercompactness, a global reflection property, and the relation of supercompactness to strong compactness. §23 describes the stronger hypotheses that evolved from Reinhardt’s proposals: extendibility and a prima facie extension shown inconsistent by Kunen. §24 then considers hypotheses on the verge of that inconsistency, and then spanning the expanse, n-hugeness and Vopěnka’s Principle. Pursuing offshoots of the theory of supercompactness, §25 describes the combinatorial study of κ γ , and §26 provides the fundamentals of extenders and related large cardinals, refined concepts that were to lead to major advances in inner model theory.

Keywords

  • Natural Sequence
  • Large Cardinal
  • Measurable Cardinal
  • Strong Compactness
  • Strong Hypothesis

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  • DOI: 10.1007/978-3-540-88867-3_6
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© 2009 Springer-Verlag Berlin Heidelberg

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(2009). Strong Hypotheses. In: The Higher Infinite. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88867-3_6

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  • DOI: https://doi.org/10.1007/978-3-540-88867-3_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-88866-6

  • Online ISBN: 978-3-540-88867-3

  • eBook Packages: Springer Book Archive