# Independence Phenomena in Mathematics:
a Set Theoretic Perspective on Current Obstacles and Scenarios for Solutions

Sandra Müller  Sandra Müller, Institut für Diskrete Mathematik und Geometrie,
TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Wien, Austria.
<EMAIL_ADDRESS>
###### Abstract.

The standard axioms of set theory, the Zermelo-Fraenkel axioms (ZFC), do not
suffice to answer all questions in mathematics. While this follows abstractly
from Kurt Gödel’s famous incompleteness theorems, we nowadays know numerous
concrete examples for such questions. A large number of problems in set
theory, for example, regularity properties such as Lebesgue measurability and
the Baire property are not decided - for even rather simple (for example,
projective) sets of reals - by ZFC. Even many problems outside of set theory
have been showed to be unsolvable, meaning neither their truth nor their
failure can be proven from ZFC. A major part of set theory is devoted to
attacking this problem by studying various extensions of ZFC and their
properties. We outline some of these extensions and explain current obstacles
in understanding their impact on the set theoretical universe together with
recent progress on these questions and future scenarios. This work is related
to the overall goal to identify the “right” axioms for mathematics.

###### Key words and phrases:

This research was funded in whole or in part by the Austrian Science Fund
(FWF) [10.55776/Y1498, 10.55776/I6087, 10.55776/V844]. For open access
purposes, the author has applied a CC BY public copyright license to any
author accepted manuscript version arising from this submission.

##  1\. Introduction

Kurt Gödel predicted in the 1930’s that the standard axioms of set theory,
Zermelo-Fraenkel set theory ($\operatorname{ZF}$) with Choice
($\operatorname{ZFC}$), do not suffice to answer all questions in mathematics.
More precisely, he showed in his first incompleteness theorem that there are
statements that can neither be proven nor disproven from the
$\operatorname{ZF}$ or $\operatorname{ZFC}$ axioms (or any sufficiently nice
extension thereof).111Gödel showed his first incompleteness theorem in fact
for an axiomatic system weaker than $\operatorname{ZF}$ but this will not be
relevant in this article. We always assume here that there is a model of
$\operatorname{ZF}$. While Gödel’s famous incompleteness theorems are abstract
theoretical results, we nowadays know numerous concrete examples for such
questions. In addition to a large number of problems in set theory, even many
problems outside of set theory are known to be unsolvable, meaning neither
their truth nor their failure can be proven from $\operatorname{ZFC}$. We call
such problems _independent from $\operatorname{ZFC}$_. This includes the
Whitehead Problem (group theory, [26]), the Borel Conjecture (measure theory,
[12]), Kaplansky’s Conjecture on Banach algebras (analysis, [4]), and the
Brown-Douglas-Fillmore Problem (operator algebras, [6]). A major part of set
theory is devoted to attacking this problem by studying various extensions of
$\operatorname{ZF}$ and their properties. One of the main goals of current
research in set theory is to identify _the “right” axioms for mathematics_
that settle these problems. This, in part philosophical, problem is attacked
with technical mathematical methods by analyzing various set theoretic
universes satisfying different extensions of $\operatorname{ZF}$.

A widely useful and well-studied collection of such axioms is given by
determinacy assumptions: These are canonical extensions of $\operatorname{ZF}$
or $\operatorname{ZFC}$ that postulate the existence of winning strategies in
natural two-player games [9, 18]. For every set of infinite sequences of
natural numbers222As common in set theory, we will tacitly identify infinite
sequences of natural numbers with reals. $A\subseteq\mathbb{N}^{\mathbb{N}}$,
we consider an infinite two-player game where player $\mathrm{I}$ and player
$\mathrm{II}$ alternate playing natural numbers $n_{0},n_{1},\dots$, as
follows:

$\begin{array}[]{c|ccccc}\mathrm{I}&n_{0}&&n_{2}&&\ldots\\\
\hline\cr\mathrm{II}&&n_{1}&&n_{3}&\ldots\end{array}$

Then player $\mathrm{I}$ wins the game if and only if the sequence
$x=(n_{0},n_{1},\dots)$ of natural numbers produced during a run of the game
is an element of $A$; otherwise, player $\mathrm{II}$ wins. We call $A$ the
_payoff set_ of this game. A set $A$ is called _determined_ if and only if one
of the two players has a winning strategy in the game with payoff set $A$,
meaning that there is a method by which they can win in the game described
above, no matter what their opponent does. The Axiom of Determinacy
($\operatorname{AD}$) is the statement that all sets of reals are determined.
Determinacy hypotheses are known to enhance sets of real numbers with a great
deal of canonical structure. But while a pioneering result of Gale and Stewart
shows that every open and every closed set in $\mathbb{N}^{\mathbb{N}}$ is
determined in $\operatorname{ZFC}$, determinacy for more complex, for example,
analytic, sets cannot be proven in $\operatorname{ZFC}$ alone. It should be
mentioned at this point that the full Axiom of Determinacy as stated above
contradicts the Axiom of Choice, so it can only be considered as an extension
of $\operatorname{ZF}$, not of $\operatorname{ZFC}$.

Other natural and well-studied extensions of $\operatorname{ZFC}$ are given by
the hierarchy of large cardinal axioms. Measurable cardinals, for example,
were introduced by Ulam [28] and further developed by Keisler and Scott [11,
25]. Determinacy assumptions, large cardinal axioms, and their consequences
are widely used and have many fruitful implications in set theory and even in
other areas of mathematics such as algebraic topology [3], topology [24, 7,
2], algebra [5], and operator algebras [6]. Many such applications, in
particular, proofs of consistency strength lower bounds, exploit the interplay
of large cardinals and determinacy axioms. Thus, understanding the connections
between determinacy assumptions and the hierarchy of large cardinals is vital
to answer questions left open by $\operatorname{ZFC}$ itself.

A third canonical line of axioms extending $\operatorname{ZFC}$ is given by
forcing axioms. They postulate that certain sets used to build canonical
extensions of the set theoretic universe exist. The first forcing axioms that
was isolated was Martin’s Axiom introduced by Martin and inspired by Solovay’s
and Tennenbaum’s proof of the independence of Souslin’s Hypothesis [15, 27].
Other well-known examples with many applications are the Proper Forcing Axiom
introduced by Baumgartner (see [16]) and the in some sense “ultimate forcing
axiom” Martin’s Maximum introduced by Foreman, Magidor, and Shelah [8].

##  2\. An example: The Continuum Problem

Arguably the most famous statement that is known to be independent from
$\operatorname{ZFC}$ is Cantor’s Continuum Problem. It was formulated by
Cantor in 1878 and appeared as the first item on Hilbert’s list of problems
announced at the International Congress of Mathematicians in Paris in 1900.
Informally, it can be phrased as the question how many real numbers there are.
Or, a bit more formally, as the following question: Is there a set $A$ of size
strictly between the size of the set of natural numbers $|\mathbb{N}|$ and the
size of the set of real numbers $|\mathbb{R}|$? I.e., is there a set $A$ such
that

$|\mathbb{N}|<|A|<|\mathbb{R}|?$

Gödel showed in the late 1930’s that there are models of set theory in which
there is no such set $A$. Many years later in 1963 Cohen was able to show that
there are also models of set theory in which there is such a set $A$, thereby
establishing the independence of the Continuum Problem from
$\operatorname{ZF}$. He was awarded the Fields Medal in 1966 for this
outstanding result.

We are interested in the question if and how canonical extensions of
$\operatorname{ZF}$ decide the Continuum Problem. Interestingly, the three
lines of extensions of $\operatorname{ZF}$ described in the introduction give
different answers to the Continuum Problem.

Determinacy:

If the Axiom of Determinacy holds, then there is no set $A$ such that
$|\mathbb{N}|<|A|<|\mathbb{R}|$.

Large Cardinals:

Large cardinal axioms do not influence the Continuum Problem, i.e., it is
still independent over $\operatorname{ZFC}$ together with the existence of
large cardinals.

Forcing Axioms:

Under the Proper Forcing Axioms there is a set $A$ such that
$|\mathbb{N}|<|A|<|\mathbb{R}|$. Even more, there is exactly one such
intermediate size between $|\mathbb{N}|$ and $|\mathbb{R}|$.

This might seem like it is the end of the dream to solve independence
phenomena in mathematics by considering canonical hierarchies of axioms
extending $\operatorname{ZF}$. In fact, this is only the beginning of a long
and fascinating line of research. Instead of just considering the direct
effect these extensions have on independent questions in mathematics, we can
analyze how “far away” the extensions are from $\operatorname{ZF}$ by
comparing their consistency strength.

##  3\. Connecting the hierarchies

Before we can describe the connections between the hierarchies of determinacy
axioms, large cardinal axioms, and forcing axioms, we outline what we mean by
a “hierarchy of axioms”. We illustrate the case for determinacy axioms as this
is the easiest to explain without further set theoretic background.

Recall that a set $A\subseteq\mathbb{N}^{\mathbb{N}}$ is called _determined_
iff one of the two players has a winning strategy in the infinite game with
payoff set $A$. The Axiom of Determinacy ($\operatorname{AD}$) postulates that
all sets of reals $A$ are determined but we can consider natural restrictions
of $\operatorname{AD}$ by narrowing the collection of determined sets. In
addition, we can consider strengthenings of $\operatorname{AD}$ by allowing
more complicated games. This leads, for example, to the following (incomplete)
list of axioms, ordered in increasing strength:

Open and closed determinacy:

Every open and every closed subset of $\mathbb{N}^{\mathbb{N}}$ is determined.

Borel determinacy:

Every Borel set $B\subseteq\mathbb{N}^{\mathbb{N}}$ is determined.

Analytic determinacy:

Every analytic set $A\subseteq\mathbb{N}^{\mathbb{N}}$ is determined. Here a
set $A$ is called _analytic_ iff it is the projection of a Borel subset of
$\mathbb{N}^{\mathbb{N}}\times\mathbb{N}^{\mathbb{N}}$ to its first
coordinate. In general, not every analytic set is Borel.

Projective determinacy:

Every projective set $A\subseteq\mathbb{N}^{\mathbb{N}}$ is determined. Here a
set $A$ is called _projective_ if it can be obtained from a Borel subset of
$(\mathbb{N}^{\mathbb{N}})^{n}$ for some $n\in\mathbb{N}$ by an iterative
alternation of taking complements and projections. By varying $n\in\mathbb{N}$
there is in fact a hierarchy of projective sets beyond the analytic sets.

Determinacy in $L(\mathbb{R})$:

Every set of reals in $L(\mathbb{R})$ is determined. Here $L(\mathbb{R})$
denotes the smallest model of $\operatorname{ZF}$ that contains all reals and
all ordinals.

Axiom of Determinacy ($\operatorname{AD}$):

Every set $A\subseteq\mathbb{N}^{\mathbb{N}}$ is determined.

Axiom of Determinacy for games on reals ($\operatorname{AD}_{\mathbb{R}}$):

Consider the infinite game where the players are allowed to play real numbers
(instead of just natural numbers). Then for every payoff set
$A\subseteq\mathbb{R}^{\mathbb{N}}$, one of the players has a winning
strategy.

Axiom of Determinacy for games of uncountable length:

Consider the infinite game where the players play uncountably many natural
numbers, i.e., the moves in the game are enumerated by ordinals
$\alpha<\omega_{1}$ (instead of just natural numbers). Then for every payoff
set $A\subseteq\mathbb{N}^{\omega_{1}}$, one of the players has a winning
strategy.

The first levels of this hierarchy are still provable in $\operatorname{ZFC}$.
It was shown by Gale and Stewart in $\operatorname{ZF}$ that every open and
every closed subset of $\mathbb{N}^{\mathbb{N}}$ is determined [9]. Here we
refer to open and closed in the product topology on $\mathbb{N}^{\mathbb{N}}$
with the discrete topology on $\mathbb{N}$. I.e., for $s\in\mathbb{N}^{n}$ for
some natural number $n$, let

$\mathcal{O}(s)=\\{x\in\mathbb{N}^{\mathbb{N}}\mid s\subset x\\},$

where $s\subset x$ for $s$ and $x$ as above denotes that $s$ is an initial
segment of $x$. Then we say that a set $A\subseteq\mathbb{N}^{\mathbb{N}}$ is
_open_ iff it is a union of such basic open sets $\mathcal{O}(s)$ for some
finite sequence of natural numbers $s$. Moreover, $A$ is _closed_ iff
$\mathbb{N}^{\mathbb{N}}\setminus A$ is open.

######  Theorem 3.1 (Gale, Stewart, 1953, [9]).

If $A\subseteq\mathbb{N}^{\mathbb{N}}$ is open or closed, then the game with
payoff set $A$ is determined.

###### Proof.

The basic idea of the proof is that membership in an open set is secured at a
finite stage and games where the winner is decided at a finite stage are
determined.

In what follows we write

$B/s=\\{x\in\mathbb{N}^{\mathbb{N}}\mid s^{\frown}x\in B\\}$

for all $B\subseteq\mathbb{N}^{\mathbb{N}}$ and $s\in\mathbb{N}^{n}$ for some
$n\in\mathbb{N}$. We will use the same notation with the obvious meaning for
$B\subseteq\mathbb{N}^{n}$ for $n\in\mathbb{N}$ as well.

######  Claim 1.

Let $s\in\mathbb{N}^{2n}$. If I does not have a winning strategy in the game
with payoff set $B/s$, then for any $i\in\mathbb{N}$, there is a
$j\in\mathbb{N}$ such that I does not have a winning strategy in the game with
payoff set $B/s^{\frown}(i,j)$.

###### Proof.

Suppose this is not the case. Let $i\in\mathbb{N}$ be a counterexample, i.e.,
for any $j\in\mathbb{N}$, I has a winning strategy $\sigma_{j}$ in the game
with payoff set $B/s^{\frown}(i,j)$.333Choosing these $\sigma_{j}$ for each
$j$ can be done in a canonical way to avoid using the Axiom of Choice. Then
the following is a winning strategy for I in the game with payoff set $B/s$:
Start by playing $i$. Then if II responds with some $j\in\mathbb{N}$, play
according to $\sigma_{j}$ for the rest of the game. ∎

Now say $A\subseteq\mathbb{N}^{\mathbb{N}}$ is open and suppose that I does
not have a winning strategy in the game with payoff set $A$. We can apply
Claim 1 recursively to obtain a strategy $\tau$ for II such that for any
partial play $s\in\bigcup_{n\in\mathbb{N}}\mathbb{N}^{2n}$ according to
$\tau$, I does not have a winning strategy in the game with payoff set $A/s$.

Suppose $\tau$ is not a winning strategy for II, i.e., there is a play $x$
according to $\tau$ such that $x\in A$. As $A$ is open, there is some
$2n\in\mathbb{N}$ such that $\mathcal{O}(x\upharpoonright 2n)\subseteq A$. But
then any strategy for I in the game with payoff set $A/(x\upharpoonright 2n)$
would be a winning strategy, contradicting the fact that $x$ is played
according to $\tau$. Therefore, $\tau$ is a winning strategy for II in the
game with payoff set $A$.

If $A\subseteq\mathbb{N}^{\mathbb{N}}$ is closed, we can use an analogous
argument with the roles of I and II interchanged. ∎

Martin showed in 1975 that also Borel determinacy can be proven in
$\operatorname{ZFC}$ [14]. But proving stronger forms of determinacy requires
extensions of $\operatorname{ZFC}$. For example, Martin showed in 1970 that,
assuming the existence of a measurable cardinal, every analytic set of reals
is determined [13]. Harrington showed a few years later that the measurable
cardinal in the hypothesis of Martin’s theorem is in some sense necessary
[10]. In fact, the following equivalence is the first level of a surprising as
well as deep connection between the hierarchies of determinacy axioms and
large cardinals.

######  Theorem 3.2 (Harrington, Martin, [10, 13]).

The following are equivalent:

  1. (1)

All analytic sets are determined, and

  2. (2)

for all $x\in\mathbb{N}^{\mathbb{N}}$, $x^{\\#}$ exists.

Defining $x^{\\#}$ (“$x$ sharp”) goes beyond the scope of this survey but we
should mention that the fact that $x^{\\#}$ exists for all
$x\in\mathbb{N}^{\mathbb{N}}$ is a consequence of the existence of a
measurable cardinal.

This deep connection between the hierarchies of determinacy axioms and large
cardinals persists throughout the projective hierarchy [21, 22, 17] and even
beyond. For example, it is known how to construct models of
$\operatorname{AD}$ or $\operatorname{AD}_{\mathbb{R}}$ from large cardinals.
But this cannot be strengthened arbitrarily: It was shown by Mycielski in the
1960’s already that the Axiom of Determinacy for games of uncountable length
is inconsistent with $\operatorname{ZF}$.

Connections between the hierarchies of large cardinals and forcing axioms
appear at a level in the large cardinal hierarchy much higher than the ones
corresponding to $\operatorname{AD}$ or $\operatorname{AD}_{\mathbb{R}}$. For
example, the existence of a supercompact cardinal implies that there is a
model of Martin’s Maximum or the Proper Forcing Axiom. In the other direction,
it is known that the Proper Forcing Axiom implies the existence of models with
certain large cardinals well below supercompact cardinals. The exact
correspondence between large cardinals and forcing axioms is a wide open
problem and has already motivated several ground breaking results in set
theory in the past decades.

##  4\. Strong axioms of determinacy

Most results obtaining so-called _large cardinal lower bounds_ for forcing
axioms, for example, constructing a model with certain large cardinal from the
Proper Forcing Axiom, exploit the deep connection between the large cardinal
hierarchy and the determinacy hierarchy. Instead of directly building models
with large cardinals, one can, for example, build a model of analytic
determinacy, and then use Theorem 3.2 to obtain large cardinal strength. One
reason why this method currently does not succeed in reaching an exact
correspondence between the Proper Forcing Axiom and some large cardinal axiom
is that the there are no known determinacy axioms that are sufficiently strong
to correspond to the levels in the large cardinal hierarchy in question. This
leads to the following question:

What are possible ways to strengthen the Axiom of Determinacy?

There are two scenarios how such a strengthening could look like.

  1. (1)

Consider restricted versions of the Axiom of Determinacy for games of
uncountable length that are still consistent with $\operatorname{ZF}$. For
example, games where the length of the game is only decided during the game
but is guaranteed to be countable or games of uncountable length with
definable payoff sets.

  2. (2)

Keep playing the same games of length $\mathbb{N}$ where the players play
natural numbers or reals but at the same time impose additional structural
properties on the models.

There are several promising results for both scenarios (see, for example, [1,
23, 19, 20]) and providing strengthenings of the Axiom of Determinacy that
correspond to high levels in the large cardinal hierarchy is among the most
important questions at the frontier of modern set theoretic research.

## References

  * [1] J. P. Aguilera and S. Müller.  The consistency strength of long projective determinacy.  J. Symb. Log., 85(1):338–366, 2020.
  * [2] R. Carroy, A. Medini, and S. Müller.  Every zero-dimensional homogeneous space is strongly homogeneous under determinacy.  J. Math. Log., 20:2050015, 2020.
  * [3] C. Casacuberta, D. Scevenels, and J. H. Smith.  Implications of large-cardinal principles in homotopical localization.  Adv. Math., 197(1):120–139, 2005.
  * [4] H. G. Dales and W. H. Woodin.  An Introduction to Independence for Analysts.  Lond. Math. Soc. Lecture Note Ser. Cambridge University Press, 1987.
  * [5] P. C. Eklof and A. H. Mekler.  Almost Free Modules, volume 65 of North-Holland Mathematical Library.  North-Holland Publishing Co., 2002.
  * [6] I. Farah.  All automorphisms of the Calkin algebra are inner.  Ann. Math., 173(2):619–661, 2011.
  * [7] W. G. Fleissner.  If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal.  Trans. Amer. Math. Soc., 273:365–373, 1982.
  * [8] M. Foreman, M. Magidor, and S. Shelah.  Martin’s maximum, saturated ideals, and nonregular ultrafilters. I.  Ann. of Math. (2), 127(1):1–47, 1988.
  * [9] D. Gale and Stewart F. M.  Infinite games with perfect information.  In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games, volume 2, pages 245–266. Princeton University Press, 1953.
  * [10] L. Harrington.  Analytic Determinacy and $0^{\\#}$.  Journal of Symbolic Logic, 43:685–693, 1978.
  * [11] H. Jerome Keisler.  Some applications of the theory of models to set theory.  Proc. Int. Cong. of Logic, Methodology, and Philosophy of Science, pages 80–85, 1962.
  * [12] R. Laver.  On the consistency of Borel’s conjecture.  Acta Math., 137:151–169, 1976.
  * [13] D. A. Martin.  Measurable cardinals and analytic games.  Fundamenta Mathematicae, 66:287–291, 1970.
  * [14] D. A. Martin.  Borel Determinacy.  Annals of Mathematics, 102(2):363–371, 1975.
  * [15] D. A. Martin and R. M. Solovay.  Internal Cohen extensions.  Ann. Math. Logic, 2(2):143–178, 1970.
  * [16] Justin Tatch Moore.  The proper forcing axiom.  In Proceedings of the International Congress of Mathematicians. Volume II, pages 3–29. Hindustan Book Agency, New Delhi, 2010.
  * [17] S. Müller, R. Schindler, and W. H. Woodin.  Mice with finitely many Woodin cardinals from optimal determinacy hypotheses.  Journal of Mathematical Logic, 20, 2020.
  * [18] J. Mycielski and H. Steinhaus.  A mathematical axiom contradicting the axiom of choice.  Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 10:1–3, 1962.
  * [19] S. Müller.  The consistency strength of determinacy when all sets are universally Baire.  2021\.
  * [20] S. Müller and G. Sargsyan.  Towards a generic absoluteness theorem for Chang models.  2023\.
  * [21] I. Neeman.  Optimal Proofs of Determinacy.  Bulletin of Symbolic Logic, 1(3):327–339, 09 1995.
  * [22] I. Neeman.  Optimal Proofs of Determinacy II.  Journal of Mathematical Logic, 2(2):227–258, 11 2002.
  * [23] I. Neeman.  The Determinacy of Long Games.  De Gruyter series in logic and its applications. Walter de Gruyter, 2004\.
  * [24] P. J. Nyikos.  A provisional solution to the normal Moore space problem.  Proc. Amer. Math. Soc., 78:429–435, 1980.
  * [25] Dana S. Scott.  Measurable cardinals and constructible sets.  Bulletin de l’Académie Polonaise des Sciences, 9:521–524, 1961\.
  * [26] S. Shelah.  Infinite abelian groups, Whitehead problem and some constructions.  Israel J. Math., 18:243–256, 1974.
  * [27] R. M. Solovay and S. Tennenbaum.  Iterated Cohen extensions and Souslin’s problem.  Ann. of Math. (2), 94:201–245, 1971.
  * [28] S. Ulam.  Zur Masstheorie in der allgemeinen Mengenlehre.  Fundamenta Mathematicae, 16:140–150, 1930.