Road to Solovay's Land.

Question

In the first semester of 2012 I took a course in General Topology and Set Theory, at undergraduate level. For topology, I was instructed to use Engelking's General Topology; albeit I had a great difficult to approach it, I got used to the text and did (and I'm still doing) some exercises (but none of the problems until now). For the Set Theory course we used Jech & Hrbacek's Introduction to Set Theory, which I think was suitable for my level back there. In these courses, I heard about Boolean Algebras, Forcing, Independence Proofs, Models, Topological Games, Kunen's book (which I just bought a copy), and others interesting things that caused me to change my favorite mathematical area (in fact I was a physics undergrad student when this year began).

In this semester, I enrolled myself in Measure and Integration course, also at undergraduate level, where I discovered about Solovay's model, which completely drove me to madness.

I'm looking for advice about my background and the path that I have to follow to reach these mentioned topics; is too early to begin? do my background is sufficiently enough to start? And where to begin with ? what books do I have to read?

P.S.: I had no background in mathematical logic, the only thing I can do is some proofs with truth-tables.

Answer 1

With no background in logic, it's a bit of a long road to Solovay's model; the good news is that every step of it is incredibly interesting!

(Some of this you may already know - I'm just listing a complete roadmap to Solovay.)

To start with, you need a good understanding of what models of ZFC look like. The last couple chapters of Hrbacek/Jech cover this; alternatively, it's at the beginning of Kunen's book.

Then comes forcing. Forcing is basically a way of building models of ZFC "to specification." This is a big deal, since ZFC is a really complicated theory, unlike, say, the theory of rings: while it's very easy to build lots and lots of (models of the theory of) rings, it's incredibly hard to build models of ZFC, and forcing accomplishes this.

The picture of forcing in ZFC is reasonably straightforward (although the details, of course, take a lot of work): you take a model V of ZFC to start with, look at some poset P in V, and the machinery of forcing gives you a* model V[G] containing V with properties that can be discovered reasonably easily by looking at P, and conversely, there are natural strategies for building a P such that the resulting V[G] will have properties you want it to. Playing around with Martin's Axiom might make forcing make a lot more sense; it certainly did for me!

(*OK, actually forcing gives you lots of different models, one for each "generic filter" G of P over V, but for almost all intents and purposes the precise generic filter doesn't matter, and all the information is contained in the poset P alone.)

Now we can prove lots of nice properties about forcing over models of ZFC, including one which for our purposes is actually a bad property: any V[G] is also a model of ZFC. The reason this is bad for us is that Solovay's model is definitely not a model of Choice, so we have to add another layer of complexity: the symmetric submodel construction. By doing some complicated shenanigans** with automorphisms of P, we can build intermediate models W of ZF set theory, containing V and contained in V[G]. Solovay's model is built in this fashion.

(**Specifically, elements of the extension V[G] have "names" in V; the symmetric submodel construction is a way of defining "hereditarily symmetric" names, which are basically names fixed by "a lot" of automorphisms of P (the precise choice of definition of "a lot" determines the properties of the symmetric submodel), and models W consist of the elements of V[G] with hereditarily symmetric names.)

So there's really four different steps in getting to Solovay's model: understanding the ZFC picture of the universe (Hrbacek/Jech's final chapters, or Kunen's intro chapter, do this well); understanding forcing over models of ZFC (Kunen covers this well, as does Jech's gigantic set theory tome); understanding symmetric submodels (this is covered in Jech's big tome, but not Kunen; so it might be a good idea to use Jech throughout); and finally, understanding the details behind Solovay's particular construction (covered in a bunch of sources, including Jech's book). Basically, Jech's giant tome of set theory - "Set Theory," Third Millennium Edition - has everything you need. It's pretty expensive, though.

Good luck!

Answer 2

Let me give an alternative ending to Noah's road map. The splitting point is at symmetric models.

After you've understood the basics of forcing well, you can switch to Kanamori's The Higher Infinite. In the chapter about the real numbers and forcing he again reviews forcing (and if you're new to this - such review is always good) and constructs Solovay's model in a very clear approach.

He avoids [1] talking about symmetric models (which can be a rather complicated tool) by using the "external" construction: we add some sort of generic set to $V$ then we consider an inner model of $V[G]$ which is $HOD(\mathbb R)$ or $L(\mathbb R)$, the latter being thrown around a lot in discussions about models of set theory without choice.

In Kanamori you can find a good introduction to large cardinals (if you haven't run into them in previous steps) which also play a role in Solovay's construction, although that appears in another chapter of the book.

I want to add that studying the construction of symmetric extensions is a good idea. This is an extremely illuminating construction which sheds a lot of light on how set theory works, at least this is how I felt in the past year. However for this particular case I think that using the approach of relative constructibility is better.

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Footnotes:

1. This is not entirely true that Kanamori avoids the symmetric models because as it turns out all symmetric models are $HOD(A)$ (whatever that means) of some generic set $A$. In the case of Solovay's model it is just much simpler to use this sort of construction rather going through the complication of symmetric forcing.