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.
