I am a first-year graduate student in maths. Around these days, I feel I must decide on which exact part of mathematics I shall go through. Infact, I have narrowed down the suitable options but still need some thoughts/advices for the matter.
Actually, I have been quite interested in model theory and for that I took an introductory type of model theory lecture in a summer kind of school. And now in my university, besides my other regular lessons like algebra, topology-geometry, every week two hours we study model theory with a post-doc student who is also a teacher in the university. So far, we covered the basics that I more or less know in a fast way and these days we are spending some time on the compactness issue(which might be the most useful thing in that). By the way, we try to cover Katrin Tent & Martin Ziegler – A Course in Model Theory. That is to say, we are on the way of doing some "hardcore" and "inside" kind of model theory. However, the pure logic side of all these also attracts my attention really so much. I remember that when I was a freshman, one of my teacher told us about Gödel's works which pretty impressed me and since then I have read(not deeply academic sources, but quite many general mathematical articles, books etc.) a lot about Gödel and, consequently, logic(beginning from Aristotle etc.). On of the most wonderful things done in mathematics(maybe not only in mathematics but ever done), to me, to my knowledge, is Gödel's Incompleteness Theorems. What he has done has always seemed such adorable to me. Thinking this way, it sounds pretty plausible to study on something around Gödel's stuff as my master thesis. Therefore, in order to be informed and take some advices, I asked one of my possible master thesis advisors in the university. He welcomed me and briefly told me about the relationship/interaction between Gödel's Incompleteness Theorems and Russell's Paradoxes. He said "it can be interesting to investigate and then present the relationship of these". This idea, as a thesis topic, totally excited me and then I immediately decided to get prepared for diving into Gödel's original paper. To this level, everything was so far so good, so clear.
However, the post-doc student that I study with warned me about this decision of thesis topic, stating that "if you are interested in logic, it is good and appreciated, you might take help from the university's philosophy department, too; but if you choose a topic which is neither pure logic nor real model theory, you will have some difficulties on continuing the way to the doctoral dissertation because there is a narrow studyground for further. Secondly, from doctoral level on, it can be really hard to find someone to study with on "logical model theory" side as there is not many people studying it and around. So to speak, in any case you have to work and get money out of it, unfortunately". Indeed, he might be right whereas I do not have much idea about what he said. All in all, I am a beginner in the academic level and can not see as much as he does for the future works. Therefore, I am confused on what to do at master level. While I was supposing that I found my area where I can combine logic and model theory, right now I am not able to see what I had better to do.
If you have any comments, opinions and advices on what I stated, they will be appreciated.
Best Answer
It is not necessary to write your Ph.D. dissertation as a direct continuation of your masters thesis. I will not write my Ph.D. as such continuation.
You could study more model theory on the side, or you could study more pure logic, or you could expand into another area. Then when the time comes to write your Ph.D. you could make a much better decision. Furthermore, I know several people who were set to solve one problem in their Ph.D. and gave up halfway only to switch to an unrelated problem.
Some universities even support external advising (especially for Ph.D. students) which means that you have a local advisor, and another advisor (often the actual advisor) to work with on your problems. You might also find it easier just to switch universities, if that's a viable option.
Besides that, it is true that it is the easiest thing to just continue your masters research into a Ph.D. dissertation, but the main use of a masters thesis is like "research training wheels" which give you a taste of doing mathematical (or otherwise) research. In the university I did my M.Sc. you are not even expected to do original research or publish papers at the end of your masters. You are only expected to write a thesis which shows that you know, a bit more, how to research a problem in mathematics.
The important thing is to do what you love. Writing a thesis, especially a good one, takes a lot of effort and time. Spending so much energy on something you dislike is not a good advice.
Let me share one experience from my masters degree. I was set to research into axiom of choice related topics, and I actually dragged my advisor into the topic. I came up with most of the questions and problems, and I made him curious about things so we studied together. Certainly if I would stay there for a Ph.D. with him we would continue to study together, even though my advisor's main interest is proper forcing, and order theory.