Pathological behavior of Borel sets?
Joel speaks on the case where the real numbers are a countable union of countable sets. The Feferman-Levy model is a strange model indeed.
However, I find the Truss construction to be even weirder. Truss repeated the construction of Solovay by starting from an arbitrary limit cardinal, and he proves that the prefect set property holds in that model, and that if we start with a singular cardinal then every set is Borel; if we start with an inaccessible cardinal then we have Solovay's model again.
Consider the Truss model when we start with $\aleph_\omega$, much like the Feferman-Levy. The difference between the two models is delicate.$\DeclareMathOperator{Col}{Col}$
Feferman-Levy begins by collapsing the $\omega_n$'s by using $\prod_{n\in\omega}\Col(\omega,\omega_n)$, and then taking sets hereditarily definable from bounded collapses.
Truss begins by collapsing all the ordinals below $\omega_\omega$ by using $\prod_{\alpha<\omega_\omega}\Col(\omega,\alpha)$, and then taking sets hereditarily definable from bounded collapses.
What is the difference? Truss allows longer and longer sequences of collapsing functions (where a collapse of an ordinal is understood as a generic for the relevant forcing), whereas Feferman-Levy only care about collapsing the cardinals.
The result is stunningly different.
In the Feferman-Levy model, the real numbers is a countable union of countable sets; in the Truss model, the countable union of countable sets of real numbers is countable.
In the Feferman-Levy model, there exists a set whose cardinality is strictly between the cardinality of the continuum and $\omega$; in the Truss model every uncountable set of reals has size continuum.
In both models, every set of real numbers is Borel.
John Truss, Models of set theory containing many perfect sets, Ann. Math. Logic 7 (1974), 197--219.
Other than that, there might be all sort of weird things going on in models of $\sf ZF$, but it's hard to say. Note that Miller's work (on Cohen's famous model) was only done recently, nearly half a century after it was first defined. Unfortunately, there's not much written work on this topic.
It is consistent with ZF that the reals $\mathbb{R}$ are a countable union of countable sets. In this case, every set of reals is Borel, and in this model one gets pathological Borel sets.
The following statement is equivalent to the perfect set property for Pi^1_1, hence not decidable in ZFC: Every sigma-compact subset of R^2 which contains uncountably many pairwise disjoint topological circles contains a subset homeomorphic to (2^omega X circle). For a proof, see my paper with Fons van Engelen and Jan van Mill, "Disjoint embeddings of compacta", Mathematika,vol. 41., 1994, 757-784.
Howard Becker