Supercompactness within the projective hierarchy (Q2747709)

In the (choice-free) ZF-context, a cardinal \(\kappa\) is called {\(\lambda\)-supercompact} if there is a fine, normal, \(\kappa\)-complete ultrafilter on \(\mathcal{P}_\kappa(\lambda)\). (Without the Axiom of Choice, this is not necessarily equivalent to the standard definition via embeddings, in particular, \(\lambda\)-supercompact cardinals can fail to be inaccessible and can actually be quite small.) NEWLINENEWLINENEWLINEA general feature of the theory under the Axiom of Determinacy is that the projective ordinals NEWLINE\[NEWLINE\pmb {\delta}^1_n:=\sup\{\alpha;\;\alpha\text{ is the length of a }\pmb{\Delta}^1_n\text{ prewellordering}\}NEWLINE\]NEWLINE have large cardinal properties. In this paper (and in this review), the base theory is \(\text{ZF}+\text{AD}\). NEWLINENEWLINENEWLINEResults of \textit{L. A. Harrington} and \textit{A. S. Kechris} [Ann. Math. Logic 20, 109-154 (1981; Zbl 0489.03018)] combined with results of \textit{D. A. Martin} and \textit{J. R. Steel} [Cabal Semin. 79-81, Proc. Caltech-UCLA Logic Semin. 1979-81, Lect. Notes Math. 1019, 86-96 (1983; Zbl 0529.03027)] yield that \(\pmb{\delta}^1_1 = \aleph_1\) is \((\pmb{\delta}^2_1)^{\mathbf{L}(\mathbb{R})}\)-supercompact and \textit{H. Becker} showed [Isr. J. Math. 40, 229-234 (1981; Zbl 0521.03036)] that \(\pmb{\delta}^1_2 = \aleph_2\) is \((\pmb{\delta}^2_1)^{\mathbf{L}(\mathbb{R})}\)-supercompact as well. NEWLINENEWLINENEWLINEIn the paper under review, the authors show that all projective ordinals are \(\lambda\)-supercompact for \(\lambda < \aleph_{\omega_1}\). NEWLINENEWLINENEWLINEThe proof consists of two parts: (a) Becker defined a notion of \(\lambda\)-regularity and showed that under certain assumptions \(\lambda\)-regular cardinals are \(\lambda\)-supercompact (Theorem 2.1). (b) Jackson proved that the projective ordinals have some degree of regularity (Theorem 4.1). NEWLINENEWLINENEWLINEBecker and Jackson combine these steps to the above result which has been superseded by a theorem of \textit{S. Jackson} from J. Symb. Log. 66, 640-657 (2001; Zbl 0988.03067), where he showed using a weak square property that the above mentioned results on \(\aleph_1\) and \(\aleph_2\) generalize to all projective ordinals, i.e., that all projective ordinals are \((\pmb{\delta}^2_1)^{\mathbf{L}(\mathbb{R})}\)-supercompact. NEWLINENEWLINENEWLINEIn addition to the interesting proofs of the two main theorems (Theorems 2.1 and 4.1), this paper presents the unpublished technique of generic codes due to Kechris and Woodin (1980) in \S\ 3.