What are the most resourceful topics to write about in set theory?

Question

I am a mathematics student in my last year of undergraduate studies and I am currently doing a project on set theory. I have so far explored Zermelo-Fraenkel Set theory with the Axiom of Choice and Elementry Theory of Category of Sets and would like to take my project further to investigate more topics. I was wondering whether there are any suggestions for specific topics or any particular topics that people have looked into and really enjoyed. I am quite new to set theory and this is the first time I've learned about it so I would really appreciate it if the answers are not too complicated.

Answer 1

One topic I quite like, which can be approached with very little background, is the study of cardinal characteristics of the continuum.

Roughly speaking (see also here), a CCC is a cardinal which measures how big a set of reals must be to be "sufficient" with respect to some relation: taking a relation $R$ between reals, or objects "morally equivalent" to reals such as functions $\mathbb{N}\rightarrow\mathbb{N}$, we can define the cardinal $$\kappa_R=\min\{\vert A\vert: \forall x\exists y\in A(yRx)\}.$$ For example:

• The actual continuum is $\kappa_{=}$.

• If we look at functions $\mathbb{N}\rightarrow\mathbb{N}$ and consider the domination relation $$fDg\iff \exists n\forall m>n[f(m)>g(m)],$$ $\kappa_D$ measures how many functions are needed to "grow as fast as every other function."

• The "dual" relation to $D$, the escaping relation $$fEg\iff \forall n\exists m>n(f(m)>g(m))\iff \neg gDf,$$ measures how many functions we need in order to be "hard to dominate." It's easy to check that $\kappa_D\ge\kappa_E$.

Note that this notation is nonstandard. In particular, $\kappa_D$ and $\kappa_E$ are properly called "$\mathfrak{d}$" and "$\mathfrak{b}$" respectively.

Trivially we always have $\kappa_R\le 2^{\aleph_0}$. Conversely, diagonal arguments can be used to show that (as long as $R$ isn't silly - and we don't call silly $\kappa_R$s CCCs) each $\kappa_R$ is uncountable. For example, given any sequence $G=(g_i)_{i\in\mathbb{N}}$ of functions $\mathbb{N}\rightarrow\mathbb{N}$, the function $$i\mapsto g(i)+1$$ is not dominated by any member of $G$; this shows that $\kappa_D$ is uncountable. A similar trick will show that $\kappa_E$ is uncountable:

Take $$i\mapsto 1+\sum_{j\le i}g(j)$$ instead.

Not all CCCs are so combinatorial in flavor. We can also consider more "analytic" ones, such as the cardinality of the smallest non-measurable set. There is a generally-agreed-upon set of ten fundamental CCCs, whose relationships are described by Cichon's diagram; verifying each of the relevant $\mathsf{ZFC}$-provable inequalities can be a good short project (if memory serves, none are particularly hard). And there are more cardinal characteristics out there, so you could continue the study of provable inequalities for quite some time.

Unfortunately (or excitingly), showing that a given inequality is not $\mathsf{ZFC}$-provable - and more generally that Cichon's diagram is complete in the relevant sense - is quite involved: we need to introduce the technique of forcing. Honestly if you're up for a project in forcing you should do that instead, perhaps tacking on a CCC-based observation at the end as a cute coda (e.g. proving the consistency of $\kappa_D>\kappa_E$). And it's worth noting that it was only recently discovered how to separate three or more CCCs simultaneously (see here).