What are the issues in modern set theory?

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

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