Regarding Shelah's approach, I believe that the following paper should be quite accessible to non-professionals: YOU CAN ENTER CANTOR’S PARADISE! Now, I have no idea how to explain Woodin's ...

By the well-known Levy-Solovay theorem, large cardinal properties are preserved under "small" forcing. Therefore CH is an assumption above ZFC which is not settled by large cardinal axioms.

" What are the advantages/disadvantages of using lower/higher-order formal logics?" It seems that "strong" logics are not as well-behaved as first order logic. By Lindstrom's theorem, FOL is the ...

Shelah has some nice results on complete extensions of first order logic. For example, he introduced cofinality quantifiers in which the truth value of a given formula (Qxy)Phi(x,y,a) is determined by ...

There are also some model-theoretic examples of such proofs (in ZFC). Once you prove that for a given logic L, the notions of consistency and completeness (for theories in L) are absolute, it's ...

It turns out that when it comes to infinite groups/modules, some algebraic concepts are deeply connected to the underlying set theory (for example, the notion of freeness, the structure of Ext, etc). ...

Pcf theory/cardinal arithmetic. Well, it's not exactly built from scratch, but there are plenty of nice results which do not use any sophisticated metamathematical machinery (such as forcing, inner ...

p < q as in "the forcing condition p is stronger than q".

The duality between measure and category in the set theory of the reals.

There is a somewhat related result by Shelah: Any Suslin ccc forcing which adds a non-dominated real adds a Cohen real. The proof can be found here: http://shelah.logic.at/files/480.pdf

In addition to Andreas' remark on the number of non-isomorphic models, perhaps it's noteworthy to say that many non-structure theorems are aiming at the construction of many models which not only are ...

I would like to extend the above question: It seems that most previous comments are dealing with independence of number theoretic statements from PA. I would like to know your input regarding the ...