Abstract
In this chapter, I develop the full theory of the strong reflecting property for L-cardinals and characterize \(\mathsf{SRP}^{L}(\omega _n)\) for \(n\in \omega \) (cf. Propositions 6.7, 6.8 and Theorem 6.2). I also generalize some results on \(\mathsf{SRP}^{L}(\gamma )\) to \(\mathsf{SRP}^{M}(\gamma )\) for other inner models M (see Theorems 6.1 and 6.4).
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Notes
- 1.
\(\overline{\overline{\gamma }}\) is the image of \(\overline{\gamma }\) under the transitive collapse of Y.
- 2.
For the definition of coherent sequences of extenders \(\overrightarrow{E}\), \(J_{\alpha }^{\overrightarrow{E}}\) and \(\overrightarrow{E}\upharpoonright \alpha \), see [3, Sect. 2.2].
- 3.
All known core models satisfy this convention.
- 4.
Here we use that \(M|\theta \) is definable in \(H_{\theta }\) for regular cardinal \(\theta >\omega _2\).
- 5.
\(\mathscr {M}(0^{\sharp }, \alpha )\) is the unique transitive \((0^{\sharp },\alpha )\)-model. For the definition of \(\mathscr {M}(0^{\sharp }, \alpha )\), I refer to Sect. 2.1.2.
- 6.
Note that \(\mathscr {M}(0^{\dag },\omega , \alpha )\) is the unique transitive \((0^{\dag },\omega ,\alpha )\)-model. For the definition of \(\mathscr {M}(0^{\dag },\omega , \alpha )\), I refer to [5].
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Cheng, Y. (2019). The Strong Reflecting Property for L-Cardinals. In: Incompleteness for Higher-Order Arithmetic. SpringerBriefs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-9949-7_6
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