I am curious about how much descriptive set theory is involved in inner model theory.
For instance Shoenfield's absoluteness result is based on the construction of the Shoenfield tree which projection is $\aleph_1$-Suslin. Also the Schoenfield tree is homogeneous, meaning the direct limit $M_x$ of the ultrapowers by the measures $\mu_{x\upharpoonright n}$ is wellfounded. The measures $\mu_{x\upharpoonright n}$ are defined on the sections of the tree. We also have that $L(\mathbb{R}) \vDash AD$ is equiconsistent with $ZFC+$ there are infinitely many Woodin cardinals. Descriptive set theory talks a lot about homogeneously Suslin sets and homogeneous trees (they pave the way for determinacy results) but these concepts seem themselves to be very important for inner model theory (just a simple fact: a set $X$ is homogeneously Suslin iff $X$ is continuously reducible to the wellfoundedness of towers of measures). The Martin Solovay tree is what gives $\Sigma^1_3$ absoluteness between $V$ and a generic extension $V[G]$ assuming measurability. Also, the Kechris-Martin Theorem has a purely descriptive theoretic proof and a purely inner model theoretic proof. A theorem of Woodin states that $(\Sigma^2_1)^{Hom_{\infty}}$ sentences are absolute for set forcing if there are arbitrarily large Woodin cardinals.
My question is why are there so many links between descriptive set theory and inner model theory? I would love to hear from an expert about the intuition as to what is really going on. The relationship between both field does not seems "ad hoc", it appears as though there is very deep beautiful and natural structure. I apologize in advance for any vagueness in my question. Thx.
Best Answer
I think that in some ways you have answered your question yourself: we see that to prove properties about sets, say within the projective hierarchy, we need representations of those sets of reals as trees, but moreover nice trees with certain properties (and the homogeneous trees you mention in particular with measures attachable to them). To get the latter involves measurable cardinals at least; and to get the determinacy of PD or AD we need to be able to shift those trees around in more subtle ways to prove that complements of nice trees, and projections of those complements etc, also have nice properties, and this requires Woodin cardinals etc...
Conversely we find descriptive set theoretic arguments involved in analysing the mouse components that go into the making of the higher inner models, in particular this is necessary for the so-called core model induction. To construct such models one has to have something of an inductive process to do so, and to start off, this involves an analysis of the lower levels of a putative model, and the description of those levels is, or can be seen as, descriptive set-theoretic on the sets of reals of the model. It just begins to look inevitable that DST and inner model theory will thus be inextricably linked. Thus the fact that the Kechris-Martin theorem has two styles of proof starts to look like two sides of the same coin.
Whilst I am not really pretending that this is a comprehensive or full answer to your question, (I hesitated to answer this as I am not setting myself up as "an expert" who you ask for!) one could add that one aspect at least, is the following reply to the (simpler, side) question that is often asked "Why large cardinals" or "Why are such large trees, and concomitantly, large sets, measures etc needed for these analyses?" It is possible to see this, at least in the determinacy arena as simply an extension of what is needed to analyse Borel: Friedman showed that we would need iterations of the power set operation (and Collection to collect together the resulting sets) with roughly speaking the number of iterations proceeding stepwise through the complexity of the Borel sets to show their determinacy (thus for $\alpha \geq \omega$ one would need $\alpha$ many iterates of power set etc to get $Det(\Sigma^0_\alpha)$). Martin's proof of Borel determinacy showed that Friedman had it exactly right: we already need larger and larger trees, or spaces if you like, in which to unravel Borel sets at a certain level and show their clopen representation (and hence determined status) in a corresponding larger space. So: we exhaust the power of ZF to show there are enough good trees to establish Borel determinacy. To get higher levels of determinacy we are going to need to go beyond ZF and use strong axioms of infinity, i.e. large cardinals.
Thus the whole enterprise spans a spectrum through building models of fragments of second order number theory second order number (up to $Det(\Sigma^0_3$) say, through models of fragments ZF of certain uncountable ordinal height (Borel Determinacy, so we use "weak" inner models of fragments of ZF of set height here), and then full height of the ordinals models of ZF, (for $Det(\Pi^1_1)$) and now comes the same with inner models for larger cardinals, albeit more sophisticated arguments.