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.
(Excerpted from an earlier version of a study guide to logic texts more generally -- you will find the latest version here: http://www.logicmatters.net/students/tyl/)
Mere lists are fairly uninteresting and unhelpful. So let's be a bit more selective!
We should certainly distinguish books covering the elements of set theory – the beginnings that anyone really ought to know about – from those that take on advanced topics such as ‘large cardinals’, proofs using forcing, etc.
On the elements, two excellent standard ‘entry level’ treatments are
Herbert B. Enderton, The Elements of Set Theory (Academic Press,
1997) is particularly clear in marking off the informal development
of the theory of sets, cardinals, ordinals etc. (guided by the
conception of sets as constructed in a cumulative hierarchy) and the
formal axiomatization of ZFC. It is also particularly good and
non-confusing about what is involved in (apparent) talk of classes
which are too big to be sets – something that can mystify beginners.
Derek Goldrei, Classic Set Theory (Chapman & Hall/CRC 1996) is
written by a staff tutor at the Open University in the UK and has the
subtitle ‘For guided independent study’. It is as you might expect
extremely clear, and is indeed very well-structured for independent
reading.
Still starting from scratch, and initially only half a notch up in sophistication, we find two more really nice books (also widely enough used to be considered "standard", whatever exactly that means):
Karel Hrbacek and Thomas Jech, Introduction to Set Theory (Marcel
Dekker, 3rd edition 1999). This goes a bit further than Enderton or
Goldrei (more so in the 3rd edition than earlier ones). The final chapter gives a remarkably accessible glimpse
ahead towards large cardinal axioms and independence proofs.
Yiannis Moschovakis, Notes on Set Theory (Springer, 2nd edition 2006). A slightly more individual path through the material than the previously books mentioned, again with glimpses ahead and again attractively written.
My next recommendation might come as a bit of surprise, as it is something of a ‘blast from the past’: but don’t ignore old classics: they can have a lot to teach us even if we have read the modern books:
- Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy, Foundations of Set-Theory (North- Holland, 2nd edition 1973). This puts the development of our canonical ZFC set theory into some context, and also discusses alternative approaches. It really is attractively readable. I’m not an enthusiast for history for history’s sake: but it is very much worth knowing the stories that unfold here.
One intriguing feature of that last book is that it doesn’t at all emphasize the ‘cumulative hierarchy’ – the picture of the universe of sets as built up in a hierarchy of levels, each level containing all the sets at previous levels plus new ones (so the levels are cumulative). This picture – nowadays familiar to every beginner – comes to the foreground again in
- Michael Potter, Set Theory and Its Philosophy (OUP, 2004). For mathematicians concerned with foundational issues this surely is – at some stage – a ‘must read’, a unique blend of mathematical exposition (mostly about the level of Enderton, with a few glimpses beyond) and extensive conceptual commentary. Potter is presenting not straight ZFC but a very attractive variant due to Dana Scott whose axioms more directly encapsulate the idea of the cumulative hierarchy of sets.
Turning now to advanced topics Two books that choose themselves as classics are
Kenneth Kunen, Set Theory (North Holland, 1980), particularly for independence proofs.
Thomas Jech, Set Theory: The Third Millenium Edition (Springer 2003), for everything.
And then there are some wonderful advanced books with narrower focus (like Bell's on Set Theory: Boolean Valued Models and Independence Proofs). But this is already long enough and in fact, if you can cope with Jech's bible, you'll be able to find your own way around the copious literature!
Best Answer
First, you've chosen a pair of terrific books, both very much suited to self-study. There are alternatives, of course, but I can't think of obviously better introductions to their respective fields.
If you have already studied, as you put it, elementary materials in logic and set theory, you'll already understand the small amount of logical notation etc. that Enderton uses, and you'll already understand the small amount of set notation etc. that Chiswell and Hodges use. So I'm sure you could proceed in either order fairly happily.
But in traditional introductory math. logic courses, it is in fact usual to cover the basics of first-order logic, as in Chiswell and Hodges, before tackling set theory, as in Enderton. And there is certainly something to be said for sticking to the traditional order. Enderton, as I recall, is rightly careful to emphasize the difference between informal set theory and formalised ZFC, and if you are to really appreciate what is going on here it will help to already be familiar with the difference between an informal and a formalised theory in the sense discussed by e.g. Chiswell and Hodges.