This is spurred by the comments to my answer here. I’m unfamiliar with set theory beyond Cohen’s proof of the independence of the continuum hypothesis from ZFC. In particular, I haven’t witnessed any real interaction between set-theoretic issues and the more conventional math I’ve studied, the sort of place where you realize “in order to really understand this problem in homotopy theory, I need to read about large cardinals.” I’ve even gotten the feeling from several professional mathematicians I’ve talked to that set theory is no longer relevant, and that if someone were to find some set-theoretic flaw in their axioms (a non-standard model or somesuch), they would just ignore it and try again with different axioms.

I also don’t personally care for the abstraction of set theory, but this is a bad reason to judge anything, especially at this early stage in my life, and I feel like I’d be more interested if I knew of some ways it interacted with the rest of the mathematical world. So:

- What do set theorists today care about?
- How does set theory interact with the rest of mathematics?
- (more subjective but) Would mathematicians working outside of set theory benefit from thinking about large cardinals, non-standard models, or their ilk?
- Could you recommend any books or papers that might convince a non-set theorist that the subject as it’s currently practiced is worth studying?
Thanks a lot!

**Answer**

Set theory today is a vibrant, active research area,

characterized by intense fundamental work both on set

theory’s own questions, arising from a deep historical

wellspring of ideas, and also on the interaction of those

ideas with other mathematical subjects. It is fascinating and I would encourage anyone to learn more about it.

Since the field is simply too vast to summarize easily,

allow me merely to describe a few of the major topics that

are actively studied in set theory today.

**Large
cardinals.**

These are the strong axioms of infinity, first studied by

Cantor, which often generalize properties true of \omega

to a larger context, while providing a robust hierarchy of

axioms increasing in consistency strength. Large cardinal

axioms often express combinatorial aspects of infinity,

which have powerful consequences, even low down. To give

one deep example, if there are sufficiently many Woodin

cardinals, then all projective sets of reals are Lebesgue

measurable, a shocking but very welcome situation. You may

recognize some of the various large cardinal

concepts—inaccessible, Mahlo, weakly compact,

indescribable, totally indescribable, unfoldable, Ramsey,

measurable, tall, strong, strongly compact, supercompact,

almost huge, huge and so on—and new large cardinal

concepts are often introduced for a particular purpose.

(For example, in recent work Thomas Johnstone and I proved

that a certain forcing axiom was exactly equiconsistent

with what we called the uplifting cardinals.) I encourage

you to follow the Wikipedia link for more information.

**Forcing.**

The subject of set theory came to maturity with the

development of forcing, an extremely flexible technique for

constructing new models of set theory from existing models.

If one has a model of set theory M, one can construct a

forcing extension M[G] by adding a new ideal element G,

which will be an M-generic filter for a forcing notion

\mathbb{P} in M, akin to a field extension in the sense

that every object in M[G] is constructible algebraically

from G and objects in M. The interaction of a model of

set theory with its forcing extensions provides an

extremely rich, intensely studied mathematical context.

**Independence Phenomenon.** The initial uses of forcing

were focused on proving diverse independence results, which

show that a statement of set theory is neither provable nor

refutable from the basic ZFC axioms. For example, the

Continuum Hypothesis is famously independent of ZFC, but we

now have thousands of examples. Although it is now the norm

for statements of infinite combinatorics to be independent,

the phenomenon is particularly interesting when it is shown

that a statement from outside set theory is independent,

and there are many prominent examples.

**Forcing Axioms.** The first forcing axioms were often

viewed as unifying combinatorial assertions that could be

proved consistent by forcing and then applied by

researchers with less knowledge of forcing. Thus, they

tended to unify much of the power of forcing in a way that

was easily employed outside the field. For example, one

sees applications of Martin’s

Axiom

undertaken by topologists or algebraists. Within set

theory, however, these axioms are a focal point, viewed as

expressing particularly robust collections of consequences,

and there is intense work on various axioms and finding

their large cardinal strength.

**Inner model
theory.**

This is a huge on-going effort to construct and understand

the canonical fine-structural inner models that may exist

for large cardinals, the analogues of Gödel’s

constructible universe L, but which may accommodate large

cardinals. Understanding these inner models amounts in a

sense to the ability to take the large cardinal concept

completely apart and then fit it together again. These

models have often provided a powerful tool for showing that

other mathematical statements have large cardinal strength.

**Cardinal characteristics of the
continuum.**

This subject is concerned with the diverse cardinal

characteristics of the continuum, such as the size of the

smallest non-Lebesgue measurable set, the additivity of the

null ideal or the cofinality of the order \omega^\omega

under eventual domination, and many others. These cardinals

are all equal to the continuum under CH, but separate into

a rich hierarchy of distinct notions when CH fails.

**Descriptive set
theory.**

This is the study of various complexity hierarchies at the

level of the reals and sets of reals.

**Borel equivalence relation
theory.**

Arising from descriptive set theory, this subject is an

exciting comparatively recent development in set theory,

which provides a precise way to understand what otherwise

might be a merely informal understanding of the comparative

difficulty of classification problems in mathematics. The

idea is that many classification problems arising in

algebra, analysis or topology turn out naturally to

correspond to equivalence relations on a standard Borel

space. These relations fit into a natural hierarchy under

the notion of Borel reducibility, and this notion provides

us with a way to say that one classification problem in

mathematics is at least as hard as or strictly harder than

another. Researchers in this area are deeply knowledgable

both about set theory and also about the subject area in

which their equivalence relations arise.

**Philosophy of set theory.** Lastly, let me also

mention the emerging subject known as the philosophy of set

theory, which is concerned with some of the philosophical

issues arising in set theoretic research, particularly in

the context of large cardinals, such as: How can we decide

when or whether to adopt new mathematical axioms? What does

it mean to say that a mathematical statement is true? In

what sense is there an intended model of the axioms of set

theory? Much of the discussion in this area weaves together

profoundly philosophical concerns with extremely technical

mathematics concerning deep features of forcing, large

cardinals and inner model theory.

**Remark.** I see in your answer to the linked question

you mentioned that you may not have been exposed to much

set theory at Harvard, and I find this a pity. I would

encourage you to look beyond any limiting perspectives you

may have encountered, and you will discover the rich,

fascinating subject of set theory. The standard

introductory level graduate texts would be Jech’s book Set

Theory and Kanamori’s book The Higher Infinite, on large

cardinals, and both of these are outstanding.

I apologize for this too-long answer…

**Attribution***Source : Link , Question Author : Paul VanKoughnett , Answer Author : JDH*