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Recently, I am learning the Kunen's set theory. Now I will reach the second part of the book, i.e., the important Martin Axiom is introduced here. I found it is a little complex and difficult for a beginner set theorist. I havn't any intuition for it. Could someone give me some simple examples? It's better to introduce some materials on Martin Axiom, such as video, ppt and so on. Thanks ahead:)

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I was in the exact same situation. Before studying forcing I tried to read about the axiom and could not understand it at all. My suggestion is to skip this part and return to it after you have studied forcing. –  Asaf Karagila Jan 15 '12 at 6:19
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Asaf, I always have my students first study Martin's axiom and its applications in order to help with the study of forcing. After all, Martin's axiom involves many of the easier issues of forcing--dense sets, filters, chain conditions, genericity--but without the complications of names and the forcing relation, which are often the most confusing for beginners. –  JDH Jan 15 '12 at 12:29
    
@Asaf: I agree very strongly with JDH: I wish that I’d encountered MA before forcing, but it was pretty new in those days. Anyone who’s already familiar with MA is free to concentrate on the really grubby technical details of forcing, because the basic idea is already clear. –  Brian M. Scott Jan 15 '12 at 16:10
    
@JDH, Brian: In a structured course I agree that it is easier, but it seems to me that John is self-studying. I tried to self-study and MA seemed awkward and somewhat unjustified (a-la Euclid's fifth postulate). Once I was comfortable with forcing and understood the whole shin-bang MA was justified and understandable. –  Asaf Karagila Jan 15 '12 at 16:34
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@Asaf: I can’t argue with your experience, but mine $-$ and I was learning it outside of formal coursework $-$ was very different: sure, MA looked a bit odd, but once I saw it used a couple of times, it was just another tool, like CH or $\diamondsuit$, harder to learn to use than the former, easier than the latter. And its topological version looks very natural to a topologist, being just a souped-up Baire category theorem. –  Brian M. Scott Jan 15 '12 at 18:10
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up vote 5 down vote accepted

If you really want to understand Martin's Axiom, read the book by Fremlin ("Consequences of Martin's Axiom"), which discusses the different variants, etc., but does not deal with proving it's consistency, using iterated forcing. IMHO, seeing the axiom used on actual problems from topology, measure theory etc. greatly enhances your understanding of it.

A more accessible approach to this, also not doing the iterated forcing, but more towards building some intuition for it, can be found in "Discovering Modern Set Theory" (part 2) by Just and Weese (the AMS bookstore sells it), especially the chapter "From the Rasiowa-Sikorski lemma to Martin's Axiom".

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Ciesielski's Set theory for the working mathematician has a chapter on Rasiowa-Sikorski lemma and Martin's Axiom, too. –  Martin Sleziak Jan 15 '12 at 8:56
    
I’ve heard good things about Fremlin’s book, but I’ve not seen it. I learned by seeing MA used and then trying to use it; once one gets the idea that the elements of the partial order are small approximations to the desired object and that the ordering itself reflects the size and mutual compatibility of these approximations, the rest is mostly practice and ingenuity in coming up with useful ccc partial orders. –  Brian M. Scott Jan 15 '12 at 16:05
    
I just ran across these notes by Don Monk; I’ve not read them in any detail, but they look fairly readable. John might find Ch. 11 on MA quite useful: it seems to be essentially a slightly gentler and less concise version of Kunen’s treatment, and it has a number of reasonable-looking exercises. –  Brian M. Scott Jan 15 '12 at 18:39
    
@Asaf;JDH;Brian;Henno; Martin:I just come back from a journey. Although I haven't read the answer and remarks carefully, thanks for your attention and useful suggestions. –  Paul Jan 16 '12 at 7:44
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