Abstract
The Boldface Martin-Harrington Theorem is the relativization of the Martin-Harrington Theorem. The former expresses that \(Det(\varvec{\varSigma _1^1})\) if and only if for any real x, \(x^{\sharp }\) exists. In this chapter, I prove the Boldface Martin-Harrington Theorem in \(\mathsf{Z_2}\) . In Sect. 3.1, I prove in \(\mathsf{Z_2}\) that if for any real \(x, x^{\sharp }\) exists, then \(Det(\varvec{\varSigma _1^1})\) holds. In Sect. 3.2, I prove in \(\mathsf{Z_2}\) that \(Det(\varvec{\varSigma _1^1})\) implies that for any real x, \(x^{\sharp }\) exists.
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Notes
- 1.
Note that for fixed real y, \(\mathsf{Z_2}+ \mathsf{HP}(y)\) implies that there is no largest L[y]-cardinal and hence L[y] is a model of \(\mathsf{ZFC}\) .
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Cheng, Y. (2019). The Boldface Martin-Harrington Theorem in \(\mathsf{Z_2}\). In: Incompleteness for Higher-Order Arithmetic. SpringerBriefs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-9949-7_3
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DOI: https://doi.org/10.1007/978-981-13-9949-7_3
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