Abstract
In his seminal paper introducing the fine structure of L, Jensen (Ann Math Log 4:229–308, 1972) proved that under \(V=L\) any regular cardinal that reflects stationary sets is weakly compact. In this paper we give a new proof of Jensen’s result that is straight-forward and accessible to those without a knowledge of Jensen’s fine structure theory. The proof here instead uses hyperfine structure, a very natural and simpler alternative to fine structure theory introduced by Friedman and Koepke (Bull Symb Log 3:453–468, 1997).
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Notes
Although the idea of the trace is implicit in the proof of the main theorem in [1] it is not isolated as a concept there.
References
Friedman, S.D., Koepke, P.: An elementary approach to the fine structure of \(L\). Bull. Symb. Log. 3(4), 453–468 (1997)
Friedman, S.D., Koepke, P., Piwinger, B.: Hyperfine structure theory and gap 1 morasses. J. Symb. Log. 71(02), 480–490 (2006)
Beller, A., Litman, A.: A strengthening of Jensen’s \(\Box \) principles. J. Symb. Log. 45, 251–264 (1980)
Silver, J.: How to eliminate fine structure from the work of Jensen. Handwritten manuscript (197?)
Jensen, R.B.: The fine structure of the constructible hierarchy. Ann. Math. Log. 4(3), 229–308 (1972)
Acknowledgements
This research was conducted under the supervision of Philip Welch with the support of an EPSRC Doctoral Training Grant Studentship, and further support from a JSPS Postdoctoral Fellowship for Research in Japan. I would like to thank an anonymous referee for detailed and valuable suggestions to improve the clarity of this article.
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Brickhill, H. Square below a non-weakly compact cardinal. Arch. Math. Logic 59, 409–426 (2020). https://doi.org/10.1007/s00153-019-00695-6
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DOI: https://doi.org/10.1007/s00153-019-00695-6