부산대학교 도서관

This work presents a set of results concerned with subforcings of the Magidor forcing, and of the Prikry forcing with non-normal ultrafilters. One can read it from a starting point of knowing measurable cardinals and the Mitchell order and the basics of forcing, and it can serve as a good introduction to the Magidor forcing. Some very interesting results are presented, and suggestions for further research questions are given at the end. \par Section 0 sets forth the content of the paper, Section 1 presents the definition of the Magidor forcing and Section 2 gives some combinatorial properties thereof. Sections 3 and 4 are concerned with proving the main result of the paper, which says that any set of ordinals $A$ that arises in one of a certain class of Magidor extensions of an arbitrary ground model $V$ is such that $V[A]$ can be recovered as $V[C']$ where $C'$ is a set of ordinals which arises as a subset of the sequence of ordinals $C_{G}$, which is the generator for the generic extension. Section 5 presents a result characterizing all complete subforcings of the Magidor forcing and Section 6 gives a result which generalizes previous results on subforcing of the Prikry forcing for normal ultrafilters to the case of $P$-point ultrafilters, while Section 7 shows that a counter-example to that result can arise in the case of non-normal ultrafilters. Section 8 presents three questions for possible future research. \par This paper is quite enjoyable to read and also basically gives a self-contained exposition of the basic properties of the Magidor forcing in the first two sections. (There was one notation used in the first section which was a bit confusing to me for a brief moment: when the definition of coherent sequence is given with the stipulation that $\alpha$ ranges over $\alpha \leq \kappa$; that's clearly restricted to measurable cardinals, and I was briefly just a bit confused over whether there was a requirement that $\alpha$ had to range over all measurable cardinals less than or equal to $\kappa$. Consultation of the references in the bibliography seems to indicate that there is indeed such a requirement, and I suppose it also becomes clear when one reads the later sections.) It's also quite appealing to look at results of a kind which actually give a complete characterization of a certain class of subforcings of a given forcing, and it's also interesting that a specific suggestion is made for a possible further result along those lines in the final section. It would be interesting to ponder whether results of that form (classifying a certain class of subforcings of a given forcing) have any further applications at all. On the whole, this paper is a good presentation of some interesting new results on sub-extensions of Magidor extensions and Prikry extensions for certain ultrafilters, and some useful ideas for further research questions.