부산대학교 도서관

This work presents a set of results concerned with subforcings of theMagidor forcing, and of the Prikry forcing with non-normalultrafilters. One can read it from a starting point of knowingmeasurable cardinals and the Mitchell order and the basics of forcing,and it can serve as a good introduction to the Magidor forcing. Somevery interesting results are presented, and suggestions for furtherresearch questions are given at the end.\par Section 0 sets forth the content of the paper, Section 1 presents thedefinition of the Magidor forcing and Section 2 gives somecombinatorial properties thereof. Sections 3 and 4 are concerned withproving the main result of the paper, which says that any set ofordinals $A$ that arises in one of a certain class of Magidorextensions of an arbitrary ground model $V$ is such that $V[A]$ can berecovered as $V[C']$ where $C'$ is a set of ordinals which arises as asubset of the sequence of ordinals $C_{G}$, which is the generator forthe generic extension. Section 5 presents a result characterizing allcomplete subforcings of the Magidor forcing and Section 6 gives aresult which generalizes previous results on subforcing of the Prikryforcing for normal ultrafilters to the case of $P$-point ultrafilters,while Section 7 shows that a counter-example to that result can arisein the case of non-normal ultrafilters. Section 8 presents threequestions for possible future research.\par This paper is quite enjoyable to read and also basically gives aself-contained exposition of the basic properties of the Magidorforcing in the first two sections. (There was one notation used in thefirst section which was a bit confusing to me for a brief moment: whenthe definition of coherent sequence is given with the stipulation that$\alpha$ ranges over $\alpha \leq \kappa$; that's clearly restricted tomeasurable cardinals, and I was briefly just a bit confused overwhether there was a requirement that $\alpha$ had to range over allmeasurable cardinals less than or equal to $\kappa$. Consultation ofthe references in the bibliography seems to indicate that there isindeed such a requirement, and I suppose it also becomes clear when onereads the later sections.) It's also quite appealing to look at resultsof a kind which actually give a complete characterization of a certainclass of subforcings of a given forcing, and it's also interesting thata specific suggestion is made for a possible further result along thoselines in the final section. It would be interesting to ponder whetherresults of that form (classifying a certain class of subforcings of agiven forcing) have any further applications at all. On the whole, thispaper is a good presentation of some interesting new results onsub-extensions of Magidor extensions and Prikry extensions for certainultrafilters, and some useful ideas for further research questions.