nameyeardelim, 75mm 75e année, 2022–2023 1207

# Strong forcing axioms and the continuum problem

following Asperó’s and Schindler’s proof that $\mathbf{MM}^{++}$ implies
Woodin’s Axiom $(*)$

Matteo Viale  Dipartimento di Matematica
Università di Torino
Via Carlo Alberto 10 - 10125
Torino - Italie<EMAIL_ADDRESS>
(Avril 2023)

## Introduction

This note addresses the continuum problem, taking advantage of the
breakthrough mentioned in the subtitle, and relating it to many recent
advances occurring in set theory.111The author acknowledges support from the
project: _PRIN 2017-2017NWTM8R Mathematical Logic: models, sets,
computability_ and from GNSAGA. We try to the best of our possibilities to
make our presentation self-contained and accessible to a general mathematical
audience.222Surveys on the topic complementing this note are (among an ample
list) [4, 19, 35, 39, 40].

Let us start by stating Asperó’s and Schindler’s result:

[[2]] Assume $\mathbf{MM}^{++}$ holds. Then Woodin’s axiom $(*)$ holds as
well.

We will address the following three questions:

  * •

What is the axiom $\mathbf{MM}^{++}$?

  * •

What is Woodin’s axiom $(*)$?

  * •

What is the bearing of Asperó’s and Schindler’s result on the continuum
problem, and why their result is regarderd as a major breakthrough in the set
theoretic community?

We give rightaway a spoiler of the type of answers we sketch for the above
questions.

We have two major approaches to produce witnesses $x$ of certain mathematical
properties $P(x)$.

A topological approach is exemplified for example by Baire’s category theorem:
given a compact Hausdorff topological space $X$ one can find a “generic” point
$x\in X$ satisfying a certain topological property $P(x)$ by showing that
$P(x)$ can fail only on a “small” (more precisely meager) set of points of
$X$.

An algebraic approach is exemplified by the construction of algebraic numbers:
one takes a set of Diophantine equations $P(\vec{x})$ which are not jointly
inconsistent, and builds abstractly a formal solution in the ring
$\mathbb{Q}(\vec{x})/_{P(\vec{x})}$.

Duality theorems connect the algebraic point of view to the geometric one, for
example Hilbert’s Nullstellensatz relates solutions of irreducible sets of
Diophantine equations to generic points of algebraic varieties.

We will outline that Woodin’s axiom $(*)$ provides an “algebraic approach” to
the construction of set theoretic witnesses for “elementary” set theoretic
properties, $\mathbf{MM}^{++}$ a “geometric approach”, and Asperó’s and
Schindler’s result connects these two perspectives.

We plan to do this while gently introducing the reader to the fundamental
concepts of set theory.

The note is structured as follows:

  * •

1 is a brief review of the basic results of set theory with a focus on its
historical development and on the topological complexity of sets of reals
witnessing the failure of the continuum hypothesis.

  * •

In 2 we quote some of Gödel’s thoughts on the continuum problem and on the
ontology of mathematical entities.

  * •

3 gives a brief overview of (the use in mathematics of) large cardinal axioms.

  * •

In 4 we introduce forcing axioms with a focus on their topological
presentations, while giving a precise formulation of the axiom
$\mathbf{MM}^{++}$. We also list some of the major undecidable problems which
get a solution assuming this axiom, among which the continuum problem.

  * •

5 is a small interlude giving a very fast insight on the forcing method, while
relating it to the notions of sheaf and of Grothendieck topos.

  * •

6 revolves about the notion of algebraic closure. In particular we outline how
Robinson’s notion of model companionship gives the means to transfer the
concept of “algebraic closure” developed for rings to a variety of other
mathematical theories.

  * •

7 discusses what is the right language in which set theory should be
axiomatized in order to unfold its “algebraic closure” properties.

  * •

8 relates Woodin’s generic absoluteness results for second order number theory
to properties of algebraic closure for the initial fragment of the universe of
sets given by $H_{\aleph_{1}}$.

  * •

9 brings to light why Woodin’s axiom $(*)$ can be regarded as an axiom of
“algebraic closure” for the larger initial fragment of set theory given by
$H_{\aleph_{2}}$. Putting everything together we conclude by showing why
Asperó’s and Schindler’s result establish a natural correspondence between the
geometric approach and the algebraic approach to forcing axioms.

I thank Alberto Albano, David Asperó, Vivina Barutello, Raphaël Carroy, Ralf
Schindler for many helpful comments on the previous drafts of this manuscript.
Many thanks to Nicolas Bourbaki for the invitation and the precious editorial
support in the preparation and revision of this work.

##  1 Basics of set theory

Set theory deals with the properties of sets (the “manageable” mathematical
objects) and classes (the “not so manageable” entities).333We refer the reader
to [20, 17, 26] for a systematic treatment of the topic. The reader familiar
with set theory can skim through or just skip this section.

###  1.1 Axioms

The axioms of set theory can be split in three types (as is the case for many
other mathematical theories):

  * •

Universal axioms which establish properties valid for all sets;

  * •

Existence axioms which establish the existence of certain sets;

  * •

Construction principles which allow for the construction of new sets from ones
which are already known to exist.

We present the axiomatization of set theory by Morse–Kelley $\mathsf{MK}$ with
sets and classes. Its axioms are distributed in the three categories as
follows:

Universal axioms

__

  * •

Extensionality: Two classes (or sets) are equal if they have exactly the same
elements.

  * •

Comprehension (a): Every class (or set) is a subset of $V$, the (proper) class
whose elements are exactly the sets.

(a proper class is a class which is not a set, a set is a class which belongs
to $V$).

  * •

Foundation: There is no infinite sequence $\langle
x_{n}:n\in\mathbb{N}\rangle$ such that $x_{n+1}\in x_{n}$ for all $n$.

Existence axioms

__

  * •

Infinity: $\emptyset$ and $\mathbb{N}$ are sets.

Weak construction principles

__

  * •

Union, Pair, Product: If $X,Y$ are sets, so are $X\cup Y$,
$\left\\{X,Y\right\\}$, $X\times Y$.

  * •

Separation: If $P$ is a class and $X$ is a set, $P\cap X$ is a set.

Strong construction principles

__

  * •

Comprehension (b): For every property $\psi(x)$, $P_{\psi}=\left\\{a\in
V:\psi(a)\right\\}$ is a class.

  * •

Replacement: If $F$ is a class function and $X\subseteq\operatorname{dom}(F)$
is a set, the pointwise image $F[X]$ of $X$ under $F$ is a set.

  * •

Powerset: If $X$ is a set, so is the class
$\mathcal{P}\left(X\right)=\left\\{Y:Y\subseteq X\right\\}.$

  * •

Global Choice: For all classes $C=\left\\{X_{i}:i\in I\right\\}$ of non-empty
sets $X_{i}$, $\prod_{i\in I}X_{i}$ (the family of functions $F$ with domain
$I$ and such that $F(i)\in X_{i}$ for all $i\in I$) is non-empty.

Some comments:

  * •

By Foundation $V$ cannot be a set else $\langle x_{n}:n\in\mathbb{N}\rangle$
with each $x_{n}$ constantly assigned to $V$ defines a decreasing
$\in$-chain.444 _$V$ is not a set_ can also be proved without Foundation. Set
theorists need foundation in order to infer that the notion of well-
foundedness is an elementary set theoretic property (more precisely it is a
provably $\Delta_{1}$-property).

  * •

Many of the objects of interest in mathematics are proper classes, for example
the family of groups, or the family of topological spaces. More generally for
a given (first order) theory $T$, the family of stuctures which satisfy the
axioms of $T$ is a proper class (and exists in view of Comprehension (b)).
There are delicate ontological issues related to the notion of proper class,
but they are foreign to almost all domains of mathematics, with the notable
exceptions of category theory and set theory.

  * •

It is convenient for natural numbers to distinguish their ordinal type (which
confronts them according to which of these numbers “comes first”) from their
cardinal type (which assigns to each natural number $n$ the family of sets
which have exactly $n$ elements). When dealing with arbitrary sets, their
ordinal type may not be defined, while the cardinal type always is. Von
Neumann devised a simple trick to represent the finite ordinal types. One can
inductively define the natural number $n$ as the set
$\left\\{0,\dots,n-1\right\\}$ (i.e. $0=\emptyset$,
$1=\left\\{\emptyset\right\\}$,
$2=\left\\{\emptyset,\left\\{\emptyset\right\\}\right\\}$, …).555The
transfinite ordinal types (or the Von Neumann ordinals) are those (possibly
infinite) sets $\alpha$ which are linearly ordered by $\in$ and are transitive
(i.e. such that when $x\in y\in\alpha$, we have that $x\in\alpha$ as well).
The proper class of Von Neumann ordinals is linearly well-ordered by $\in$.
One can check that the natural numbers are the finite Von Neumann ordinals and
that $\mathbb{N}$ is the first infinite Von Neumann ordinal.

  * •

Set theoretic construction principles are of two sorts: the simple (or weak)
ones are for example those bringing from sets $X,Y$ to sets $X\cup Y$,
$\left\\{X,Y\right\\}$, $X\times Y$, or from set $X$ and class $P$ to the set
$P\cap X$; the strong ones are the power-set axiom, the replacement axiom, and
the axiom of choice. Let us discuss briefly the role of such axioms in the
development of routine mathematics.

Weak construction principles

The integers and rationals can be constructed from the naturals using only
weak construction principles: $\mathbb{Z}$ can be seen as the subset of
$\mathbb{N}\times\left\\{0,1\right\\}$ which assigns the positive integers to
the ordered pairs with second coordinate $0$ and the negative ones to those
pairs with second coordinate $1$ (paying attention to the double counting of
$0$ as $(0,0)$ and $(0,1)$); $\mathbb{Q}$ can be seen as the subset of
$\mathbb{Z}\times(\mathbb{N}\setminus\left\\{0\right\\})$ given by ordered
pairs which are coprime.

Powerset axiom

In order to build the reals from the rationals, one needs this axiom:
$\mathbb{R}$ is the subset of $\mathcal{P}\left(\mathbb{Q}\right)$ given by
Dedekind cuts.

Replacement axiom

An adequate development of set theory requires it: consider the function $F$
on $\mathbb{N}$ given by $F(0)=\mathbb{N}$,
$F(n+1)=\mathcal{P}\left(F(n)\right)$. Without replacement it cannot be proved
that $F$ (or even the image of $F$) is a set, it might only be a proper class.

Choice

Choice also has a special status in ordinary mathematics, and many
mathematicians feel uneasy about it. However Choice is unavoidable: it is
essential in the proofs of the Hahn–Banach theorem, of the existence of a base
for infinite-dimensional vector spaces, or of the existence of a maximal ideal
on a ring,…Even the equivalence of sequential continuity and topological
continuity for real valued functions requires it: if
$f:\mathbb{R}\to\mathbb{R}$ is not continuous at $x$, there is $\epsilon>0$
such that for each $n$ one can find $x_{n}$ so that $|x_{n}-x|<1/n$ and
$|f(x_{n})-f(x)|>\epsilon$. The sequence $(x_{n})_{n}$ is (and in most cases
can only be) defined appealing to (countable) Choice.

###  1.2 Cardinal arithmetic

Let us now develop arithmetic on cardinal types.

Given sets $X$, $Y$

  * •

The cardinal $|X|$ is the (proper) class $\left\\{Y:\,\exists\,f:X\to Y\text{
bijection}\right\\}$;

  * •

$|X|\leq|Y|$ if and only if _there is $f:X\to Y$ injection_ if and only if
_there is $g:Y\to X$ surjection_;666The last equivalence gives an alternative
definition of the axiom of choice.

  * •

$|X|<|Y|$ iff $|X|\leq|Y|$ and $|X|\neq|Y|$;

  * •

$|X|+|Y|=|(X\times\left\\{0\right\\})\cup(Y\times\left\\{1\right\\})|$ (the
size of the disjoint union of $X$ and $Y$);

  * •

$|X|\cdot|Y|=|X\times Y|$ (the size of the product of $X$ and $Y$);

  * •

$|X|^{|Y|}=|X^{Y}|$ where

$X^{Y}=\left\\{f:\text{ $f$ is a function with domain $Y$ and range
$X$}\right\\}.$

Note that each cardinality class on a set $X$ is a proper class if it contains
a non-empty element. The equivalence class of $\emptyset$ is
$\left\\{\emptyset\right\\}$. It can also be shown that $C$ is a proper class
if and only if there is a surjection of $C$ onto $V$, and $X$ is a set if and
only if there is no surjection of $X$ onto $V$.

The finite sets are those in the equivalence class of some $n\in\mathbb{N}$.
For finite sets $X,Y$ of size respectively $m,n$:

  * •

$|X|=|Y|$ if and only if $n=m$ if and only if $|X|\leq|Y|$ and $|Y|\leq|X|$;

  * •

$|X|+|Y|$ is (in the equivalence class of) $m+n$;

  * •

$|X|\cdot|Y|$ is (in the equivalence class of) $m\cdot n$;

  * •

$|X|^{|Y|}$ is (in the equivalence class of) $m^{n}$;

i.e. the map $n\mapsto|n|$ defines an embedding of the stucture
$(\mathbb{N},<,\cdot,+,n\mapsto 2^{n},0,1)$ in the family of cardinals endowed
by the above operations.

Basic questions were then asked for the infinite cardinals:

  1. 1.

Assume $|X|\leq|Y|$ and $|Y|\leq|X|$ holds for infinite sets $X,Y$, do we have
also in this case that $|X|=|Y|$?

  2. 2.

What is the structure of the order on infinite cardinals given by $<$?

  3. 3.

Do the arithmetic of cardinals obey the associative, commutative, distributive
laws which hold for the naturals?

  4. 4.

What is the computation table for the arithmetic operations of sum and product
on cardinals, when one of the factors is an infinite cardinal?

  5. 5.

What about the computation table for the exponential map $|X|\mapsto 2^{|X|}$
when $X$ is infinite?

Soon most of these questions had a clear cut answer:

  1. 1.

For arbitrary sets $X,Y$ we have that $|X|\leq|Y|$ and $|Y|\leq|X|$ if and
only if $|X|=|Y|$ (Cantor 1887, Bernstein 1897, Dedekind 1898).777This is non
trivial: there is a topological embedding of $[0;1]$ into $(0;1)$ and
conversely; there is no continuous bijection of $[0;1]$ onto $(0;1)$; however
one can find a Borel bijection.

  2. 2.

$\leq$ is a well-order on cardinals (Zermelo+…$\sim$ 1904), i.e. it is a
linear order on cardinals such that for every class $C\neq\emptyset$ there is
$\min\left\\{|X|:X\in C\right\\}$.888 Furthermore in each class $|X|$ there is
a least Von Neumann ordinal $\kappa$ which is called the cardinal of $X$ and
is the canonical representative of $|X|$.

  3. 3.

The arithmetic operations on cardinals endow this family of a structure of
ordered commutative semiring with exponentiation; the natural numbers form an
initial segment of this semiring (Cantor - published in 1895, but most likely
known earlier).

  4. 4.

$|X|+|Y|=|X|\cdot|Y|=\max\left\\{|X|,|Y|\right\\}$ if at least one among $X,Y$
is infinite and both are non-empty (Special cases by Cantor, before 1895,
general proofs by Harward 1905, Jourdain 1908, Hausdorff 1914…).

  5. 5.

For all sets $X$, $|X|<|\mathcal{P}\left(X\right)|=|2|^{|X|}$ (Cantor
1891).999$x\mapsto\left\\{x\right\\}$ witnesses
$|X|\leq|\mathcal{P}\left(X\right)|$; if $g:\mathcal{P}\left(X\right)\to X$,
$Y_{g}=\left\\{y\in Y:y\not\in g(y)\right\\}$ witnesses that $g$ is not a
surjection. Regarding the map $(|X|,|Y|)\mapsto|X|^{|Y|}$, the computation of
its table can be done in terms of that of the exponential map [6].

We denote cardinals by greek letters $\kappa,\lambda,\theta$.

  * •

$\aleph_{0}$ is the cardinality of $\mathbb{N}$ the least infinite cardinal;

  * •

for $\kappa$ a cardinal $\kappa^{+}$ is the cardinality of the least cardinal
above $\kappa$ (which exists by 2 and 5);

  * •

$\aleph_{1}=\aleph_{0}^{+}$, $\aleph_{2}=\aleph_{1}^{+}$.

###  1.3 The continuum problem

The so called continuum problem (Cantor 1878) remained unsettled:

> What is the value of $2^{|\mathbb{N}|}$?

The continuum hypothesis can be equivalently phrased:

  * •

$2^{\aleph_{0}}=\aleph_{1}$;

  * •

For all $X\subseteq\mathbb{R}$ either $X$ is countable or $X$ has size
continuum.

The equivalence of the two follows once one observes that
$|\mathbb{R}|=|\mathcal{P}\left(\mathbb{N}\right)|$.

###  1.4 Progresses on the definable version of the continuum problem

For decades there were scant progresses on the solution of this problem. On
the positive side one can note proofs that “simply definable” subsets of
$\mathbb{R}$ cannot witness the negation of the continuum hypothesis (the so
called “continuum hypothesis for definable sets”).

The _analytic_ (or $\Sigma^{1}_{1}$) subsets of $\mathbb{R}^{n}$ are the
projections on the first $n$-coordinates of a Borel subset of
$\mathbb{R}^{n+k}$ for some natural number $k$. The _coanalytic_ (or
$\Pi^{1}_{1}$) subsets of $\mathbb{R}^{n}$ are the complements of the analytic
subsets.

  * •

No _closed_ subset of $\mathbb{R}$ is a counterexample to $\mathsf{CH}$
(Cantor 1883).

  * •

No _Borel_ subset of $\mathbb{R}$ is a counterexample to $\mathsf{CH}$
(Alexandroff 1916, Hausdorff 1917).

  * •

No _analytic_ subset of $\mathbb{R}$ is a counterexample to $\mathsf{CH}$
(Suslin+Alexandroff 1917).

  * •

_Coanalytic_ subsets of $\mathbb{R}$ have either size
$\aleph_{0},\aleph_{1},2^{\aleph_{0}}$ (Luzin–Sierpinski 1917$\approx$).

This was as far as the first rounds of mathematicians addressing Cantor’s
problem could get.

With the introduction of large cardinals further major progresses were
obtained on the definable version of $\mathsf{CH}$.

The _projective_ subsets of $\mathbb{R}^{n}$ are those subsets of
$\mathbb{R}^{n}$ which are $\Sigma^{1}_{m}$ (or $\Pi^{1}_{m}$) for some $m$,
where $X\subseteq\mathbb{R}^{n}$ is $\Sigma^{1}_{m+1}$ if it is the projection
of a $\Pi^{1}_{m}$-subset of $\mathbb{R}^{n+k}$ and is a
$\Pi^{1}_{m+1}$-subset if it is the complement of a $\Sigma^{1}_{m+1}$-set.

Projective sets define a natural family of subsets of $\mathbb{R}^{n}$ which
ought to be topologically simple.

Another natural family has been isolated by [13]:

Recall that for a given topological space $(Y,\tau)$, a set $X$ is _nowhere
dense_ in $Y$ if its complement contains an open dense subset of $Y$, it is
_meager_ if it is a countable union of nowhere dense subsets, it has the
_Baire property_ if it has meager symmetric difference with an open set.
[[13]] $X\subseteq\mathbb{R}^{k}$ is _universally Baire_ if for all continuous
maps $f:Y\to\mathbb{R}^{k}$ with $Y$ compact Hausdorff, $f^{-1}[X]$ has the
Baire property in $Y$. Analytic sets are universally Baire, and this family
forms a $\sigma$-algebra.

To appreciate the strength of this property, consider $2^{\mathbb{N}}$ when
$2$ is given the discrete topology and $2^{\mathbb{N}}$ has the product
topology.

The map $\theta:f\mapsto\sum_{i=0}^{\infty}f(i)/3^{i+1}$ for
$f:\mathbb{N}\to\left\\{0,2\right\\}$ defines a topological embedding of
$2^{\mathbb{N}}$ into $[0;1]$ whose image is meager and has Lebesgue measure
$0$.

Now take a subset $P$ of $2^{\mathbb{N}}$ which does not have the Baire
property in $2^{\mathbb{N}}$.

Seen as a subset of $[0;1]$, $\theta[P]$ is meager, hence it has the Baire
property in $[0;1]$. On the other hand $P=\theta^{-1}[\theta[P]]$ does not
have the Baire property in $2^{\mathbb{N}}$. Hence $\theta[P]$ is not
universally Baire even if it has the Baire property.

A similar argument based on measure, shows that the non-measurable sets
produced by Vitali are not universally Baire. More generally it can be shown
that assuming large cardinals a subset of the reals is “non-pathological” if
and only if it is universally Baire. The theorem below makes this a sound
mathematical assertion. Assume there is a proper class of Woodin cardinals.
Then:

  * •

No universally Baire subset of $\mathbb{R}$ is a counterexample to
$\mathsf{CH}$ [13, 24, 11].

  * •

Borel sets, analytic sets, projective sets,…are all universally Baire [13].

  * •

The universally Baire subsets of $\mathbb{R}$ are determined [24].

  * •

The determined subsets of the reals are Lebesgue measurable, have the perfect
set property,…(various authors one for each relevant regularity property).

###  1.5 Independence of $\mathsf{CH}$

$\mathsf{CH}$ is independent of the axioms of set theory:

  * •

There is a model of the axioms of $\mathsf{MK}$ where $\mathsf{CH}$ holds
[15].

  * •

There is a model of the axioms of $\mathsf{MK}$ where $\mathsf{CH}$ fails [9].

  * •

In the model of the axioms of $\mathsf{MK}$ where $\mathsf{CH}$ fails produced
by Cohen, this failure can be witnessed by a $\Sigma^{1}_{2}$-set.

##  2 Gödel’s program

[15] wrote an influential survey on the continuum problem which has been a
source of inspiration for the work of many logicians to come. In my opinion
the best summary of its content can be given by quoting a few excerpts from
the paper:

On the independence of the continuum problem (p. 520)

_Only someone who (like the intuitionist) denies that the concepts and axioms
of classical set theory have any meaning (or any well-defined meaning) could
be satisfied with such a solution, not someone who believes them to describe
some well-determined reality. For in this reality Cantor’s conjecture must be
either true or false, and its undecidability from the axioms as known today
can only mean that these axioms do not contain a complete description of this
reality;_

On Large Cardinals (p. 520)

_For first of all the axioms of set theory by no means form a system closed in
itself, but, quite on the contrary, the very concept of set on which they are
based suggests their extension by new axioms which assert the existence of
still further iterations of the operation “set of”. These axioms can also be
formulated as propositions asserting the existence of very great cardinal
numbers or (which is the same) of sets having these cardinal numbers. The
simplest of these strong “axioms of infinity” assert the existence of
inaccessible numbers (and of numbers inaccessible in the stronger sense)
$>\aleph_{0}$._

On success as a criterion to detect new axioms (p. 521)

_There might exist axioms so abundant in their verifiable consequences,
shedding so much light upon a whole discipline, and furnishing such powerful
methods for solving given problems (and even solving them, as far as that is
possible, in a constructivistic way) that quite irrespective of their
intrinsic necessity they would have to be assumed at least in the same sense
as any well established physical theory._

##  3 Large cardinals

Consider the universe of sets construed from the emptyset using all other
axioms with the exception of _Infinity_ ; it can be shown that one ends up
with a “baby” universe $H_{\aleph_{0}}$ characterized by the following
property: if $X\in H_{\aleph_{0}}$, $X$ is finite, all the elements $Y$ of $X$
are finite, all the elements $Z$ of some element $Y$ of $X$ are finite,…More
precisely Given a set $X$,

  * •

$\cup^{0}X=X$, $\cup^{n+1}X=\cup(\cup^{n}X)$,

  * •

$\operatorname{trcl}(X)=\bigcup_{n\in\mathbb{N}}(\cup^{n}X)$ is the
_transitive closure_ of $X$.

$X$ is _hereditarily finite_ if $\operatorname{trcl}(X)$ is finite.

$H_{\aleph_{0}}$ is the collection of hereditarily finite sets.

One can check that $\mathcal{P}\left(H_{\aleph_{0}}\right)$ (or -more
precisely- the structure
$(\mathcal{P}\left(H_{\aleph_{0}}\right),H_{\aleph_{0}},\in)$) is a model of
all axioms of Morse–Kelley set theory with the exception of the infinity axiom
asserting that $\mathbb{N}$ is a set. Indeed in this model $H_{\aleph_{0}}$ is
the class of all sets and $\mathbb{N}\subseteq H_{\aleph_{0}}$ is a “proper
class”.

In particular the axiom of Infinity is a way to assert that not all sets can
be described using the construction principles and starting from the emptyset.
$\mathbb{N}$ is an example of such a set.

Large cardinal axioms posit the existence of sets which cannot be described
from $\mathbb{N}$ using the construction principles encoded in the axioms of
$\mathsf{MK}$. We already mentioned from Gödel the axiom stating the existence
of inaccessible cardinals, i.e. cardinals behaving like $|\mathbb{N}|$, but of
larger size: $\kappa$ is inaccessible if and only if
$\mathcal{P}\left(H_{\kappa}\right)$ is a model of Morse–Kelley set theory
whose universe of sets is $H_{\kappa}$,101010Equivalently $\kappa$ is
inaccessible if it is regular and $(H_{\kappa},\in)\models\mathsf{ZFC}$.
where: Given a cardinal $\lambda$, a set $X$ is _hereditarily of size less
than $\lambda$_ if $\operatorname{trcl}(X)$ has size less than $\lambda$.

$H_{\lambda}$ is the set of all sets which are hereditarily of size less than
$\lambda$.

The original proof by Wiles of Fermat’s last theorem uses the notion of
Grothendieck universe; when correctly formalized in set theory, the existence
of a Grothendieck universe is equivalent to the existence of an inaccessible
cardinal. Grothendieck’s theory of universes finds its natural formulation in
(it is actually equivalent to) set theory enriched with the axiom stating the
existence of a proper class of inaccessible cardinals [25].

Wiles’ proof provides evidence grounded on Gödel’s criterion of success for
the adoption of large cardinal axioms. There are plenty of (less celebrated)
such cases. For example:

[Vopenka’s principle $\mathsf{VP}$] For every _proper class_ of directed
graphs with no loops, there are two members of the class with a homomorphism
between them. This is an equivalent formulation of $\mathsf{VP}$ by [1].
$\mathsf{VP}$ is a tool which category theorists employ successfully, see for
example [5, 7, 29].

$\mathsf{VP}$ entails the existence of all large cardinal axioms one may
require in any of the results presented elsewhere in this note; for example if
$\mathsf{VP}$ holds there is a proper class of supercompact cardinals;
furthermore any supercompact cardinal is also Woodin. Henceforth assuming
$\mathsf{VP}$ no counterexample to $\mathsf{CH}$ can be universally Baire, the
projective sets of reals are universally Baire, the universally Baire sets are
determined.

Later on we will delve more on the effects of large cardinals on second order
numebr theory and on the family of topologically simple sets of reals.

##  4 Forcing axioms

Loosely speaking forcing axioms try to encapsulate the idea that the powerset
of some set $X$ is “as thick as possible”. Forcing axioms for $X$ can be
divided in two categories:

Axioms of topological maximality

They can be reformulated as strong forms of Baire’s category theorem, are
inspired by the notion of generic point, include $\text{{\sf MM}}^{++}$ among
their instantiations.

Axioms of algebraic maximality

They assert the closure of $\mathcal{P}\left(X\right)$ under a variety of set
theoretic operations, are inspired by the notion of algebraic closure, include
Woodin’s axiom $(*)$ among their instationations.

We remark the following:

  * •

$\text{{\sf MM}}^{++}$ and $(*)$ are forcing axioms for $X=\aleph_{1}$.

  * •

Baire’s category theorem is a “topological” forcing axiom for
$X=\mathbb{N}=\aleph_{0}$.

  * •

Large cardinals entail “algebraic” forcing axioms for
$X=\mathbb{N}=\aleph_{0}$.

###  4.1 Topological maximality and Martin’s maximum

Recall Baire’s category theorem:

> Let $(X,\tau)$ be a compact Hausdorff space and
> $\left\\{D_{n}:n\in\mathbb{N}\right\\}$ be a countable family of dense open
> subsets of $X$. Then $\bigcap_{n}D_{n}$ is dense in $X$.

Let us parametrize the conclusion in all cardinals $\kappa$ rather than just
$\aleph_{0}$: Let $\kappa$ be an infinite cardinal and $(X,\tau)$ a
topological space.

$\mathsf{FA}_{\kappa}(X,\tau)$ holds if $\bigcap_{i\in\kappa}D_{i}$ is _dense_
in $X$ for all $\left\\{D_{i}:\,i\in\kappa\right\\}$ family of dense open
subsets of $X$.

For $\kappa>\aleph_{0}$ not all $(X,\tau)$ compact Hausdorff satisfy
$\text{{\sf FA}}_{\kappa}(X,\tau)$. For example:

> Let $Y$ be an _uncountable set_ and $(X,\tau)$ be the Stone-Čech
> compactification of the product space $Y^{\mathbb{N}}$ where $Y$ is endowed
> of the discrete topology. Then $\text{{\sf FA}}_{\aleph_{1}}(X,\tau)$ fails.

It can also be shown that $\text{{\sf FA}}_{\aleph_{1}}(X,\tau)$ holds for
certain compact Hausdorff spaces $(X,\tau)$.111111Actually an equivalent
formulation of the axiom of choice states that $\text{{\sf
FA}}_{\kappa}(X,\tau)$ holds for any compact Hausdorff space $(X,\tau)$ such
that $\tau$ admits a base $P$ of non-empty sets with the property that
$(P,\subseteq)$ is a $<\kappa$-closed forcing (see for example [36]).

Abraham isolated a necessary condition on $(X,\tau)$ so that
$\mathsf{FA}_{\aleph_{1}}(X,\tau)$ is not inconsistent.

######  Proposition 1 (Abraham).

Assume $(X,\tau)$ is a compact Hausdorff space which is not121212See Def. 9;
however now we do not need to know what SSP means for now. $\mathsf{SSP}$.
Then $\text{{\sf FA}}_{\aleph_{1}}(X,\tau)$ fails.

[14] showed that it can also be a sufficient condition:

[[14]] Martin’s maximum MM $\equiv$_ $\text{{\sf FA}}_{\aleph_{1}}(X,\tau)$
holds for all compact Hausdorff spaces $(X,\tau)$ which are $\mathsf{SSP}$._

[[14]] Assume there exists a proper class of supercompact cardinals. Then
there is a model of $\mathsf{MK}$ where MM holds and there is a proper class
of supercompact cardinals.

In particular the theorem establishes that there is a model of set theory
enriched with large cardinal axioms such that

> _ $\text{{\sf FA}}_{\aleph_{1}}(X,\tau)$ holds for all compact Hausdorff
> spaces $(X,\tau)$ for which it is not impossible._

###  4.2 What are the mathematical consequences of forcing axioms?

We collect here some of the major applications of forcing axioms in set theory
and in other domains of mathematics.

Assume Martin’s maximum holds. Then:

  * •

$\mathsf{CH}$ is false and the continuum is the second uncountable cardinal,
i.e. $2^{\aleph_{0}}=\aleph_{2}$ [14].

  * •

Whitehead’s conjecture on free groups is false, (i.e. there are uncountable
Whitehead groups which are not free) [30].

  * •

Kaplansky’s conjecture on Banach algebras holds (i.e. every algebra
homomorphism from the Banach algebra $C(X)$ -where $X$ is compact Hausdorff-
into any other Banach algebra is necessarily continuous) (Woodin and Solovay,
unpublished).131313A complete proof from published sources can be obtained
combining results in [10, Chapter 3] with [32, Thm. 7.7].

  * •

There are five uncountable linear orders such that any uncountable linear
order contains an isomorphic copy of one of them [27]

  * •

All automorphisms of the Calkin algebra are inner [12].

  * •

…

All these conclusions are independent of $\mathsf{MK}+$_Vopenka’s principle_
(or any other large cardinal assumption).

###  4.3 $\mathsf{MM}^{++}$

$\mathsf{MM}^{++}$ is a natural technical strengthening of $\mathsf{MM}$. For
the sake of completeness we state one of its possible definitions; we caution
the reader that it will make sense only for those familiar with the forcing
method and with the basic theory of $\mathcal{P}\left(\aleph_{1}\right)$.
Those not willing to delve into set theoretic technicalities may skip to the
next section. This axiom also appears in [14], as well as all the results of
this section, unless otherwise specified.

Given a complete boolean algebra $\mathsf{B}$, $\dot{S}\in V^{\mathsf{B}}$ is
a $\mathsf{B}$-name for a stationary subset of $\aleph_{1}$, if
$\left\llbracket\dot{S}\text{ is a stationary subset of
}\dot{\aleph}_{1}\right\rrbracket_{\mathsf{B}}=1_{\mathsf{B}}.$

Given $G$ ultrafilter on $\mathsf{B}$, we let
$\dot{S}_{G}=\left\\{\alpha<\aleph_{1}:\left\llbracket\check{\alpha}\in\dot{G}\right\rrbracket_{\mathsf{B}}\in
G\right\\}.$

Given $C$ club subset of $\aleph_{1}$, we let
$D_{C,\dot{S}}=\left\\{G\in\operatorname{St}(\mathsf{B}):\dot{S}_{G}\cap
C\neq\emptyset\right\\}.$

Note that $D_{C,\dot{S}}$ is dense open in $\operatorname{St}(\mathsf{B})$ for
any $C$ club subset of $\aleph_{1}$.

Recall that $(X,\tau)$ is a compact extremally disconnected Hausdorff space if
and only if it is the Stone space of its algebra of regular open sets.

######  Proposition 2.

For a compact Hausdorff space $(X,\tau)$, $\text{{\sf
FA}}_{\aleph_{1}}(X,\tau)$ holds if and only if so does $\text{{\sf
FA}}_{\aleph_{1}}(\operatorname{St}(\mathsf{B}),\tau_{\mathsf{B}})$; where
$\mathsf{B}$ is the algebra of regular open subsets of $X$ and
$\tau_{\mathsf{B}}$ is the compact Hausdorff topology on the Stone space
$\operatorname{St}(\mathsf{B})$ of ultrafilters on $\mathsf{B}$ induced by the
Stone duality applied to $\mathsf{B}$.

Given a compact extremally disconnected Hausdorff space $(X,\tau)$, we let
$\mathsf{B}$ be the complete boolean algebra given by its regular open sets.

$\text{{\sf FA}}^{++}_{\aleph_{1}}(X,\tau)$ holds if for any family
$\left\\{D_{\alpha}:\alpha<\aleph_{1}\right\\}$ of dense open subsets of $X$,
and any family $\left\\{\dot{S}_{\alpha}:\alpha<\aleph_{1}\right\\}$ of
$\mathsf{B}$-names for stationary subsets of $\aleph_{1}$, there is a
ultrafilter $G\in X=\operatorname{St}(\mathsf{B})$ such that

$G\in D_{\alpha}\text{ for all }\alpha<\aleph_{1},$

and

$G\in D_{C,\dot{S}_{\alpha}}\text{ for all }\alpha<\aleph_{1}\text{ and $C$
club subset of $\aleph_{1}$}.$

Note that $2^{\aleph_{1}}>\aleph_{1}$ and there are $2^{\aleph_{1}}$-many
clubs $C$ on the cardinal $\aleph_{1}$, each one producing a different set
$D_{C,\dot{S}_{\alpha}}$ for each $\alpha<\aleph_{1}$. Therefore $\text{{\sf
FA}}^{++}_{\aleph_{1}}(X,\tau)$ requires the existence of points on $X$
meeting families of dense open sets of size much larger than $\aleph_{1}$.

$\mathsf{MM}^{++}$ holds if $\text{{\sf FA}}^{++}_{\aleph_{1}}(X,\tau)$ holds
for all compact extremally disconnected Hausdorff spaces which are
$\mathsf{SSP}$.

It is known (for example by a combination of [2, 22]) that $\mathsf{MM}^{++}$
is an axiom strictly stronger than $\mathsf{MM}$. However the standard proof
of the consistency of Martin’s maximum produces a model of $\mathsf{MM}^{++}$,
and we have the following:

Assume there is a proper class of supercompact cardinals. Then there is a
model of $\mathsf{MK}$ with a proper class of supercompact cardinals where
$\text{{\sf MM}}^{++}$ holds.

##  5 Forcing

We give in this section some basic information on the forcing method and its
relation with the notion of Grothendieck Topos. The content of this section is
not needed in the subsequent parts of this note.

We recall that Stone duality identifies complete boolean algebras with compact
Hausdorff extremally disconnected spaces (a space is extremally disconnected
if the closure of an open set is open).

Below we organized a text sketching on its left-side column the forcing
procedure according to a set theorist, and highlighting on the right-side
column the corresponding steps viewed with the lenses of a category theorist.
The reader can skim through the left-side text, then through the right side
text, and finally compare the paragraphs of the two texts with same alignment.

2

SET THEORIST

CATEGORY THEORIST

Given the complete boolean algebra $\mathsf{B}$,

Given the compact extremally disconnected Hausdorff space
$\operatorname{St}(\mathsf{B})$,

one forms

2 the boolean valued model $V^{\mathsf{B}}$
by Cohen/Scott–Solovay–Vopenka; the topos
$\mathsf{Sh}(\operatorname{St}(\mathsf{B}),\mathsf{CompHaus})$, given by
topological sheaves on $\operatorname{St}(\mathsf{B})$ with target a compact
Hausdorff space;

one chooses

2 a $V$-generic ultrafilter $G$ on $\mathsf{B}$, a generic point of
$\operatorname{St}(\mathsf{B})$ belonging to all dense open subset of
$\operatorname{St}(\mathsf{B})$,

and one obtains

2 the $\mathsf{MK}$-model $V[G]$ generic extension of $V$ by $G$. the topos
$\mathsf{Sh}(\operatorname{St}(\mathsf{B}),\mathsf{CompHaus})/_{G}$ of stalks
at $G$ of the sheaves in
$\mathsf{Sh}(\operatorname{St}(\mathsf{B}),\mathsf{CompHaus})$.

The properties of

2 $V[G]$ depend mainly on $\mathsf{B}$
$\mathsf{Sh}(\operatorname{St}(\mathsf{B}),\mathsf{CompHaus})/_{G}$ depend
mainly on $\operatorname{St}(\mathsf{B})$

and minimally on $G$.141414Note that generic points do not exist for atomless
boolean algebras, however it can be of help for our intuition to work under
the assumption that such points exist and that -on average- the points of
$\operatorname{St}(\mathsf{B})$ behave like generic points. A strictly formal
approach to the topic can eliminate the use of generic points in the forcing
analysis of the properties of $V^{\mathsf{B}}$ (or equivalently of
$\mathsf{Sh}(\operatorname{St}(\mathsf{B}),\mathsf{CompHaus})$).

We give some glimpses on how forcing assigns truth values to logical
properties in a structure of the form $\mathcal{F}(X)/_{G}$ when $\mathcal{F}$
is a sheaf on a compact extremally disconnected Hausdorff space $X$, and $G$
is a point of this space. The machinery of forcing generalizes to the topos of
such sheaves this procedure.

Consider the space $L^{\infty+}(\mathbb{R})$ of measurable functions
$f:\mathbb{R}\to\mathbb{R}\cup\left\\{\infty\right\\}$ which take value
$\infty$ on a null set. For example $1/x$ is such a function and all
essentially bounded measurable functions are also in
$L^{\infty+}(\mathbb{R})$. Note that with pointwise multiplication and sum
$L^{\infty+}(\mathbb{R})/_{\approx}$ is a commutative ring (where $\approx$
identifies functions overlapping an a conull set, and $f+g(x)=f\cdot g(x)=0$
if either one of $f(x),g(x)$ is $\infty$).

Let $\mathsf{MALG}$ be the complete boolean algebra given by Lebesgue
measurable sets modulo null sets. Given a universally Baire (hence Lebesgue
measurable) relation $R\subseteq\mathbb{R}^{n}$ we can lift it to a boolean
relation

$R^{\mathsf{MALG}}:L^{\infty+}(\mathbb{R})^{n}\to\mathsf{MALG}$

by setting

$R^{\mathsf{MALG}}(f_{1},\dots,f_{n})=[\left\\{x\in\mathbb{R}:R(f_{1}(x),\dots,f_{n}(x))\right\\}],$

where $[A]$ is the equivalence class in $\mathsf{MALG}$ of the measurable set
$A\subseteq\mathbb{R}$. For example

$\sin(x)<^{\mathsf{MALG}}\cos(x)=[\bigcup_{n\in\mathbb{Z}}((2n-1)\cdot\pi+\frac{\pi}{4};2n\cdot\pi+\frac{\pi}{4})]$

gets a $\mathsf{MALG}$-value which is neither true nor false.

A key observation is that whenever $R,S\subseteq\mathbb{R}^{n}$ are
universally Baire (Lebesgue measurable), so are $R\cap S,R\cup
S,\mathbb{R}^{n}\setminus S$. Furthermore (assuming the existence of a proper
class of Woodin cardinals) so is also $\pi_{j}[R]$ where $\pi_{j}$ is the
projection on the $j$-th coordinate of $R$.

This gives that

$\displaystyle(R\cap S)^{\mathsf{MALG}}(f_{1},\dots,f_{n})$ $\displaystyle=$
$\displaystyle[\left\\{x\in\mathbb{R}:R(f_{1}(x),\dots,f_{n}(x))\right\\}\cap\left\\{x\in\mathbb{R}:S(f_{1}(x),\dots,f_{n}(x))\right\\}]$
$\displaystyle=$ $\displaystyle R^{\mathsf{MALG}}(f_{1},\dots,f_{n})\wedge
S^{\mathsf{MALG}}(f_{1},\dots,f_{n})$

Similarly it can be shown (appealing again to universal Baireness) that:

$(\mathbb{R}^{n}\setminus
S)^{\mathsf{MALG}}(f_{1},\dots,f_{n})=\neg_{\mathsf{MALG}}S^{\mathsf{MALG}}(f_{1},\dots,f_{n}),$
$(R\cup
S)^{\mathsf{MALG}}(f_{1},\dots,f_{n})=R^{\mathsf{MALG}}(f_{1},\dots,f_{n})\vee
S^{\mathsf{MALG}}(f_{1},\dots,f_{n}),$
$(\pi_{n}[R])^{\mathsf{MALG}}(f_{1},\dots,f_{n-1})=\bigvee_{\mathsf{MALG}}\left\\{R^{\mathsf{MALG}}(f_{1},\dots,f_{n-1},g):g\in
L^{\infty+}(\mathbb{R})\right\\}.$

Now when $G$ is a ultrafilter on $\mathsf{MALG}$, we can pass to the quotient
structure $L^{\infty+}(\mathbb{R})/_{G}$ given by the classes

$[f]_{G}=\left\\{h:[f=h]\in G\right\\}$

and relations

$R/_{G}([f_{1}]_{G},\dots,[f_{n}]_{G})$

holding when $R^{\mathsf{MALG}}(f_{1},\dots,f_{n})\in G$.

One can check (among many things) that the axioms of ordered field holds in
$L^{\infty+}(\mathbb{R})/_{G}$, as $G$ makes a coherent selection of truth
values. For example given $f,g,h\in L^{\infty+}(\mathbb{R})$ it cannot be the
case that the three boolean values
$(f<^{\mathsf{MALG}}g),(g<^{\mathsf{MALG}}h),(h<^{\mathsf{MALG}}f)$ are all
simultaneously in $G$. Similar arguments yield that the relation
$<^{\mathsf{MALG}}/_{G}$ defines a dense linear order without endpoints on
$L^{\infty+}(\mathbb{R})/_{G}$.

Letting $f\approx g$ iff $[\left\\{x:f(x)=g(x)\right\\}]=1_{\mathsf{MALG}}$,
$L^{\infty+}(\mathbb{R})/_{\approx}$ is the space of global sections with
regard to the sheafification according to the dense Grothendieck topology of
the presheaf $\mathcal{F}$ which to any $[A]\in\mathsf{MALG}$ assigns the
family of essentially bounded measurable functions defined on $A$ modulo null
sets. $L^{\infty+}(\mathbb{R})/_{G}$ is then the stalk at $G$ of this space of
global sections (see [28] for details on this example).

Cohen’s forcing method applied to $\mathsf{MALG}$ devises a procedure which
generalizes the above simultaneously to the whole topos
$\mathsf{Sh}(\operatorname{St}(\mathsf{MALG}),\mathsf{CompHaus})$ (which in
forcing is described as the structure $V^{\mathsf{MALG}}$) producing a
$\mathsf{MALG}$-valued model of set theory. The procedure is modular and can
be applied with input any complete boolean algebra $\mathsf{B}$ (or any
extremally disconnected compact Hausdorff space), and links the logical
properties of the $\mathsf{B}$-valued structure $V^{\mathsf{B}}$ to the
combinatorial properties of $\mathsf{B}$. By suitably choosing $\mathsf{B}$,
Cohen was able to produce a $\mathsf{B}$-valued model of set theory where
$\mathsf{CH}$ gets value $0_{\mathsf{B}}$, or (equivalently) such that
$\mathsf{CH}$ is false in $V^{\mathsf{B}}/_{G}$.

##  6 Algebraic maximality and model companionship

It is now time to make a detour in model theory and investigate the notion of
algebraic closure and its possible transfers/generalizations to other
mathematical theories.151515Standard model theory textbooks are [8, 16, 23,
31].

###  6.1 Algebraic closure of structures

Consider the signature $\left\\{+,\cdot,0,1\right\\}$. The table below shows
on the left column the algebraic theory under consideration, in the central
column the first order axiomatization in the above signature, in the right
column a standard model of the axioms.

Structures | Axioms | Example
---|---|---
|  |
| $\forall x,y\,(x\cdot y=y\cdot x)$ |
| $\forall x,y,z\,[(x\cdot y)\cdot z=x\cdot(y\cdot z)]$ |
Commutative | $\forall x\,(x\cdot 1=x\wedge 1\cdot x=x)$ |
semirings | $\forall x,y\,(x+y=y+x)$ |
with no zero | $\forall x,y,z\,[(x+y)+z=x+(y+z)]$ | $\mathbb{N}$
divisors | $\forall y\,(x+0=x\wedge 0+x=x)$ |
| $\forall x,y,z\,[(x+y)\cdot z=(x\cdot y)+(x\cdot z)]$ |
| $\forall x,y\,[x\cdot y=0\rightarrow(x=0\vee y=0)]$ |
|  |
|  |
Integral |  |
domains | $\forall x\exists y\,(x+y=0)$ | $\mathbb{Z}$
|  |
|  |
Fields | $\forall x\,[x\neq 0\rightarrow\exists y\,(x\cdot y=1)]$ | $\mathbb{Q}$
|  |
|  |
Algebraically | $\text{for all }n\geq 1$ |
closed fields | $\forall x_{0}\dots x_{n}\exists y\,\sum x_{i}\cdot y^{i}=0$ | $\mathbb{C}$
|  |

The table describes some features of the passages from $\mathbb{N}$ to
$\mathbb{Z}$, from $\mathbb{Z}$ to $\mathbb{Q}$, from $\mathbb{Q}$ to
$\mathbb{C}$. On the semantic level these passages came along with the
introduction of new operations, and the construction of structures closed
under these operations (additive inverses, multiplicative inverses, closure
under solutions of polynomial equations), examples of which are listed in the
left-side column. On the syntactic side the cell detected by the intersection
of the first row with the central column lists the universal axioms common to
all theories; the closure under the new operations of the structures on the
left column is reflected by the satisfaction of corresponding
$\Pi_{2}$-axioms, as listed in the corresponding cells of the central column.

Roughly the more closed-off a semiring with no zero-divisors is, the more
$\Pi_{2}$-axioms it satisfies.

Robinson came up with model theoretic notions giving an abstract description
of the closed-off structures for an arbitrary universal theory.

###  6.2 Existentially closed structures and model companionship

  * •

A vocabulary $\tau$ is a list of predicates, constants and function symbols
(we are accustomed to the vocabulary $\left\\{+,\cdot,0,1,<,=\right\\}$ which
we can use to write down diophantine (in)equations, such as $x+2y<xz+5w$).
Associated to a vocabulary $\tau$ we naturally have a notion of atomic
$\tau$-formula (which for $\left\\{+,\cdot,0,1,<,=\right\\}$ are exactly the
diophantine (in)equations).

  * •

A $\tau$-formula $\phi(x_{1},\dots,x_{n})$ is _quantifier free_ if it is a
boolean combination of _atomic_ formulae (i.e. obtained by atomic formulae
using conjunction $\wedge$, disjunction $\vee$, negation $\neg$, implication
$\to$); for example $(x+y=1+1+z)\wedge\neg(x\cdot y<z\cdot z)$ is quantifer
free for $\tau=\left\\{\cdot,+,0,1,<\right\\}$.

  * •

A $\tau$-formula $\psi(x_{1},\dots,x_{n})$ is a $\Sigma_{1}$-formula if it is
of the form

$\exists x_{0},\dots,x_{k}\,\phi(x_{0},\dots,x_{k},x_{k+1},\dots,x_{n})$

with $\phi(x_{0},\dots,x_{k},x_{k+1},\dots,x_{n})$ quantifier free; for
example

$\exists x,y\,[(x+y=1+1+z)\wedge\neg(x\cdot y<z\cdot z)]$

is $\Sigma_{1}$ for $\tau=\left\\{\cdot,+,0,1,<\right\\}$).

In the vocabulary $\left\\{+,\cdot,0,1\right\\}$, the atomic formulae are
_diophantine equations_ and the _quantifier free formulae_ with parameters in
a ring $\mathcal{M}$ define the _constructible sets_ (in the sense of
algebraic geometry) of $\mathcal{M}$. Below a standard example of a quantifier
free formula (with parameters in $\mathcal{M}$):

$\bigvee_{j=1}^{l}\left[\bigwedge_{i=1}^{k_{j}}p_{ij}(a^{ij}_{1},\dots,a^{ij}_{m_{ij}},x_{1},\dots,x_{n})=0\wedge\bigwedge_{d=1}^{m_{j}}\neg
q_{dj}(b^{dj}_{1},\dots,b^{dj}_{k_{dj}},x_{1},\dots,x_{n})=0\right]$

with $a^{ij}_{k},b^{dj}_{k}$ elements of $\mathcal{M}$ and
$p_{ij}(y_{1},\dots,y_{m_{ij}},x_{1},\dots,x_{n})$,
$q_{dj}(z_{1},\dots,z_{k_{dj}},x_{1},\dots,x_{n})$ polynomials with
coefficients in $\mathbb{N}$ (of degree $1$ in the $y_{l},z_{h}$-s).

Given a vocabulary $\tau$ and $\tau$-structures161616 $\sqsubseteq$ denotes
the substructure relation among the $\tau$-structures
$\mathcal{M}\sqsubseteq\mathcal{N}$, $\mathcal{M}\prec_{1}\mathcal{N}$ if
every $\Sigma_{1}$-formula with parameters in $\mathcal{M}$ and true in
$\mathcal{N}$ is true also in $\mathcal{M}$. For example
$\langle\mathbb{C},+,\cdot,0,1\rangle{\prec_{1}}\langle\mathbb{C}[X],+,\cdot,0,1\rangle,$
but
$\langle\mathbb{Z},+,\cdot,0,1\rangle{\not\prec_{1}}\langle\mathbb{C},+,\cdot,0,1\rangle$.

Given a $\tau$-theory $S$, a $\tau$-structure $\mathcal{M}$ is $S$-ec if:

  * •

there is a model of $S$ $\mathcal{N}\sqsupseteq\mathcal{M}$,

  * •

$\mathcal{M}\prec_{1}\mathcal{N}$ for any $\mathcal{N}\sqsupseteq\mathcal{M}$
which models $S$.

For $S$ the $\left\\{+,\cdot,0,1\right\\}$-theory of _integral domains_ the
_algebraically closed fields_ are exactly the $S$-ec models.

Given a $\tau$-theory $S$, a $\tau$-theory $T$ is the _model companion_ of $S$
if TFAE for any $\tau$-structure $\mathcal{M}$:

  * •

$\mathcal{M}$ is a model of $T$,

  * •

$\mathcal{M}$ is $S$-ec.

The $\left\\{+,\cdot,0,1\right\\}$-theory of _integral domains_ has the
$\left\\{+,\cdot,0,1\right\\}$-theory of _algebraically closed fields_ as its
model companion.

A $\tau$-theory $T$ is _model complete_ if it is its own model companion, i.e.
if $\mathcal{M}\prec_{1}\mathcal{N}$
whenever171717$\mathcal{M}\prec\mathcal{N}$ holds if all formulae with
parameters in $\mathcal{M}$ get the same truth value in the two structures
(hence $\prec_{1}$ is weaker than $\prec$). Model completeness can
equivalently be defined replacing $\prec_{1}$ by $\prec$.
$\mathcal{M}\sqsubseteq\mathcal{N}$ are models of $T$.

The $\left\\{+,\cdot,0,1\right\\}$-theory of _algebraically closed fields_ is
_model complete_.

In particular model companionship and model completeness describe a notion of
algebraic closure which makes sense for an arbitrary mathematical theory, when
axiomatized in a certain signature $\tau$.

Note that the signature plays a crucial role in model companionship results;
for example the theory of algebraically closed fields is the model companion
of the theory of integral domains in signature
$\left\\{+,\cdot,0,1\right\\}$.181818It is also the model companion of the
theory of fields in this same signature. It is not anymore so in the signature
$\left\\{+,\cdot,^{-1},0,1\right\\}$ where to interpret -1 in an integral
domain we use the axiom:

$\forall x\,[\exists y\,(x\cdot y=1\wedge x\cdot x^{-1}=1)\vee(\neg\exists
y\,(x\cdot y=1)\wedge x^{-1}=0)].$

With this axiom we still get that

$\mathcal{M}=\langle\mathbb{C},+,\cdot,^{-1},0,1\rangle\mathrel{\sqsubseteq}\mathcal{N}=\langle\mathbb{C}[X],+,\cdot,^{-1},0,1\rangle,$

but not that $\mathcal{M}\mathrel{\prec_{1}}\mathcal{N}:$ $\exists x\,(x\neq
0\wedge x^{-1}=0)$ is true in $\mathcal{N}$ (as witnessed by $X$), but false
in $\mathcal{M}$.

In particular $\langle\mathbb{C},+,\cdot,^{-1},0,1\rangle$ is not anymore
$T$-ec for $T$ the theory of integral domains in signature
$\left\\{+,\cdot,^{-1},0,1\right\\}$. Note however that the theory of
algebraically closed fields is still model complete191919It is also still the
model companion of the theory of fields in
$\left\\{+,\cdot,^{-1},0,1\right\\}$. even when formalized in signature
$\left\\{+,\cdot,^{-1},0,1\right\\}$.

##  7 Algebraic closure for set theory

We now want to use the notion of model companionship to describe what is
“algebraic closure” for set theory. This will allow to formulate a reasonably
accessible definition of Woodin’s axiom $(*)$.

###  7.1 The right vocabulary for set theory

The standard axiomatization of set theory in textbooks is done in vocabulary
$\left\\{\in\right\\}$, eventually with extra symbol $\subseteq$. However this
is not giving an efficient formalization. For example in the
$\left\\{\in\right\\}$-vocabulary the notion of ordered pair is formalized by
means of _Kuratowski’s trick:_ $\langle y,z\rangle$ is (coded by) the set
$\left\\{\left\\{y\right\\},\left\\{y,z\right\\}\right\\}$.

The standard $\in$-formula expressing $x=\langle y,z\rangle$ is

$\exists t\exists u\;[\forall w\,(w\in x\leftrightarrow w=t\vee
w=u)\wedge\forall v\,(v\in t\leftrightarrow v=y)\wedge\forall v\,(v\in
u\leftrightarrow v=y\vee v=z)].$

We do not regard the concept of ordered pair as a complicated one. It should
then be formalizable by a simple formula. Accordingly many other basic set
theoretic concepts such as that of being an $n$-ary relation, a function, the
emptyset, the set of natural numbers,…should all be formalizable by simple
formulae.

The lightface $\Delta_{0}$-properties isolate the simplest set theoretic
properties and include the following classes:

  * •

$\left\\{R\in V:R\text{ is an $n$-ary relation}\right\\}$,

  * •

$\left\\{f\in V:f\text{ is a function}\right\\}$,

  * •

$\left\\{\langle a,b\rangle:a\subseteq b\text{ are sets}\right\\}$,

  * •

…

[20, Def. IV.3.5]

An $\left\\{\in\right\\}$-formula $\phi$ is a $\Delta_{0}$-formula if all its
quantified variables are bounded to range in a set (e.g. $y\subseteq
z\,\equiv\,\forall x\,(x\in y\rightarrow x\in z)\,\equiv\,\forall x\in
y\,(x\in z)$ is a $\Delta_{0}$-formula).

A lightface $\Delta_{0}$-property is a class of sets whose extension is given
by a $\Delta_{0}$-formula.

It is thus natural to expand the vocabulary of set theory as follows:

__

$\in_{\Delta_{0}}$ has the following list of symbols:

  * •

constants for $\emptyset,\mathbb{N}$,

  * •

relation symbols $R_{\phi}$ for any $\Delta_{0}$-formula
$\phi(x_{1},\dots,x_{n})$,

  * •

function symbols $G_{i}$ for the list of ten Gödel operations given in [17,
Def. 13.6].

The base $\in_{\Delta_{0}}$-theory for sets interprets the new symbols
according to their expected meaning, for example it must include the following
axioms:

  * •

$\forall\vec{x}\,[R_{\phi}(\vec{x})\leftrightarrow\phi(\vec{x})]$ whenever
$\phi$ is a $\Delta_{0}$-formula

  * •

$\forall\vec{x},y\,(R_{\phi_{i}}(\vec{x},y)\leftrightarrow G_{i}(\vec{x})=y)$
if $\phi_{i}(\vec{x},y)$ is the $\Delta_{0}$-formula describing the graph of
the Gödel operation $G_{i}$.

  * •

$\forall x\,(x\not\in\emptyset)$.

  * •

$\forall x\,(x\in\mathbb{N}\leftrightarrow x\text{ is a finite Von Neumann
ordinal})$.

Note that $(x\text{ is a finite Von Neumann ordinal})$ is (formalizable by) a
$\Delta_{0}$-formula.

We now give an explicit first order axiomatization of $\mathsf{MK}$ in
signature $\in_{\Delta_{0}}\cup\left\\{\mathsf{Set},V\right\\}$ where
$\mathsf{Set}$ is a unary predicate symbol for the property of being a set and
$V$ is a constant which denotes the universe of all sets. We stipulate for the
sake of readability that smallcase letters indicate sets, uppercase letters
indicate classes. To the above list of $\in_{\Delta_{0}}$-axioms202020To be
picky, the above list of axioms should be reformulated so that quantifiers
apply only to variables ranging over sets. we add:

Universal axioms

__

Extensionality

$\forall X,Y\,[(X\subseteq Y\wedge Y\subseteq X)\leftrightarrow X=Y]$.

Comprehension (a)

$\forall X\,(\mathsf{Set}(X)\leftrightarrow X\in V)\wedge\forall
X\,(X\subseteq V)$.

Foundation

$\forall F\,[(F\text{ is a function
}\wedge\operatorname{dom}(F)=\mathbb{N})\,\rightarrow\exists
n\in\mathbb{N}\,F(n+1)\not\in F(n)].$

Existence Axioms

__

Emptyset

$\mathsf{Set}(\emptyset)$,

Infinity

$\mathsf{Set}(\mathbb{N})$.

Basic construction principles

__

Gödel operations

For each $i=1,\dots,10$

$\forall x_{0},\dots,x_{k_{i}}[(\bigwedge_{i=0}^{k}(x_{i}\in
V)\rightarrow\exists!z\,(z\in V\wedge R_{\phi_{i}}(\vec{x},z))],$

where $\phi_{i}$ is the $\Delta_{0}$-formula whose extension is the graph of
the Gödel operation $G_{i}$.

Separation

$\forall P,x\,[\mathsf{Set}(x)\rightarrow\mathsf{Set}(P\cap x)]$.

Strong construction principles

__

Comprehension (b)

For every $\in_{\Delta_{0}}$-formula $\psi(x_{0},\dots,x_{n},\vec{Y})$

$\forall\vec{Y}\,\exists Z\,\forall x\,[x\in Z\leftrightarrow(x\in
V\wedge\exists x_{0},\dots,x_{n}(x=\langle
x_{0},\dots,x_{n}\rangle\wedge\psi(x_{0},\dots,x_{n},\vec{Y})))].$

Replacement

$\forall F,x\,[(F\text{ is a function
}\wedge\mathsf{Set}(x)\wedge(x\subseteq\operatorname{dom}(F)))\rightarrow\mathsf{Set}(F[x])].$

Powerset

$\forall
x\,[\mathsf{Set}(x)\rightarrow\exists!y\,[\mathsf{Set}(y)\wedge(\forall
z\,(z\in y\leftrightarrow z\subseteq x))]].$

Global Choice

$\displaystyle\forall F[$ $\displaystyle F\text{ is a function}\wedge\forall
x\,(x\in\operatorname{dom}(F)\rightarrow F(x)\neq\emptyset)$
$\displaystyle\rightarrow$ $\displaystyle\exists G\,(G\text{ is a
function}\wedge\operatorname{dom}(G)=\operatorname{dom}(F)\wedge\forall
x\,(x\in\operatorname{dom}(G)\rightarrow G(x)\in F(x))$ $\displaystyle]$

An $\in_{\Delta_{0}}$ model of $\mathsf{MK}$ is a “sorted” structure
$(\mathcal{C},V,\in_{\Delta_{0}})$, where $\mathcal{C}$ is the family of all
classes, and $V$ is an element of $\mathcal{C}$ whose extension is the
subfamily of $\mathcal{C}$ given by sets. One can also check that in this case
$(V,\in_{\Delta_{0}})$ is a model of the $\mathsf{ZFC}$-axioms (the classical
presentation of set theory which avoids the mention of proper classes).

With a certain degree of approximation, it is customary to denote a model
$(\mathcal{C},V,\in_{\Delta_{0}})$ by its second component $V$, as
$\mathcal{C}$ could be recovered as the family of subclasses of $V$. However
this is not entirely correct as there can be
$(\mathcal{C},V,\in_{\Delta_{0}})$, $(\mathcal{D},V,\in_{\Delta_{0}})$ both
models of $\mathsf{MK}$ with $\mathcal{C}\neq\mathcal{D}$. When confusion on
this issue may arise we will be more explicit. In general we stick to the
convention to denote a model of (some of the axioms of) $\mathsf{MK}$ by
$(\mathcal{C},V,\in_{\Delta_{0}})$ and when we write just
$(V,\in_{\Delta_{0}})$ we are considering the substructure whose elements are
sets and not proper classes.

###  7.2 The $H_{\kappa}$s

A finite set may not be simple, for example to understand the singleton
$\left\\{\mathbb{R}\right\\}$ we need to know $\mathbb{R}$. We want to
stratify sets according to the cardinalities required to generate them. Recall
(Def. 3 and Def. 3) that a set $X$ is _hereditarily of size less than (at
most) $\kappa$_ if $\operatorname{trcl}(X)$ has size less than (at most)
$\kappa$.

  * •

$\left\\{\mathbb{R}\right\\}$ is not hereditarily countable (i.e.
$\left\\{\mathbb{R}\right\\}\notin H_{\aleph_{1}}$);

  * •

Any subset $A$ of $\mathbb{N}$ is hereditarily countable (i.e. $A\in
H_{\aleph_{1}}$);

  * •

$\mathbb{Q}$ and $\mathbb{Z}$ as defined in any textbook are hereditarily
countable (i.e. in $H_{\aleph_{1}}$);

  * •

$\mathbb{R}$ and $\mathcal{P}\left(\mathbb{N}\right)$ are subsets of
$H_{\aleph_{1}}$ (but not elements!);

  * •

$\mathcal{P}\left(\mathbb{N}\right)$ is definable by the atomic
$\in_{\Delta_{0}}$-formula $(x\subseteq\mathbb{N})$ in the structure $\langle
H_{\aleph_{1}},\in_{\Delta_{0}}\rangle$;

  * •

similarly for $\mathbb{R}$ or for (a representative of the homeomorphism class
of) any Polish space.

  * •

$\mathcal{P}\left(\aleph_{1}\right)$ is definable by the atomic
$\in_{\Delta_{0}}$-formula $(x\subseteq\aleph_{1})$ in parameter $\aleph_{1}$
(the first uncountable ordinal) in the structure $\langle
H_{\aleph_{2}},\in_{\Delta_{0}}\rangle$,

  * •

$\mathsf{NS}$, the non-stationary ideal on $\aleph_{1}$, is
$\Sigma_{1}$-definable in parameter $\aleph_{1}$ in the same structure (see
Section 9 for a definition of $\mathsf{NS}$).

$H_{\aleph_{0}}\subseteq H_{\aleph_{1}}\subseteq
H_{\aleph_{2}}\subseteq\dots\subseteq H_{\kappa^{+}}\subseteq\dots$

$V=\bigcup\left\\{H_{\lambda}:\lambda\text{ an infinite cardinal}\right\\}$

$(\mathcal{P}\left(H_{\aleph_{i}}\right),H_{\aleph_{i}},\in_{\Delta_{0}})$ for
$i=1,2$ are models of all axioms of $\mathsf{MK}$ with the exception of
powerset. The role of $V$ in these models is played by
$H_{\aleph_{i}}$.212121Similarly it can be shown that for $i=1,2$,
$(H_{\aleph_{i}},\in_{\Delta_{0}})$ are model of all axioms of $\mathsf{ZFC}$
with the exception of Powerset.

$(\mathcal{P}\left(H_{\kappa}\right),H_{\kappa},\in_{\Delta_{0}})$ models
$\mathsf{MK}$ if and only if $\kappa$ is inaccessible.

###  7.3 Existentially closed structures for set theory

[Levy absoluteness] Let $\kappa$ be an infinite cardinal.

Then

$\langle
H_{\kappa^{+}},\in_{\Delta_{0}},A:A\subseteq\mathcal{P}\left(\kappa\right)\rangle\prec_{1}\langle
V,\in_{\Delta_{0}},A:A\subseteq\mathcal{P}\left(\kappa\right)\rangle$

We argue below that $H_{\aleph_{1}}$ and $H_{\aleph_{2}}$ provide natural
candidates for existentially closed structures for set theory. The choice of
whether to focus on $H_{\aleph_{1}}$ or $H_{\aleph_{2}}$ depends on the
signature in which one formalizes set theory.

##  8 Algebraic maximality for $\mathcal{P}\left(\mathbb{N}\right)$ and
generic absoluteness

We connect here the notion of algebraic closure given by existentially closed
model, to generic absoluteness results for second order arithmetic.

[Shoenfield, 1961] Let $V[G]$ be a forcing extension of $V$. Then222222Note
that $\langle H_{\aleph_{1}}^{V[G]},\in_{\Delta_{0}}\rangle\prec_{1}\langle
V[G],\in_{\Delta_{0}}\rangle$ follows from Levy Absoluteness applied in
$V[G]$.

$\langle H_{\aleph_{1}},\in_{\Delta_{0}}\rangle\prec_{1}\langle
H_{\aleph_{1}}^{V[G]},\in_{\Delta_{0}}\rangle.$

A key observation is that $\mathcal{P}\left(\mathbb{N}\right)$ can be
identified with Cantor’s space $2^{\mathbb{N}}$ via characteristic functions.
With this identification the universally Baire subsets232323Def. 1.4 was given
only on $\mathbb{R}^{k}$ but makes sense for any uncountable Polish space $X$
(a second countable topological space whose topology can be induced by a
complete metric) which is locally compact. Actually the families of
universally Baire subsets of uncountable locally compact Polish spaces $X$ and
$Y$ can be identified modulo any Borel isomorphism existing between the two
spaces [13, 18]. of $2^{\mathbb{N}}$ describe a particularly nice
$\sigma$-algebra contained in
$\mathcal{P}\left(\mathcal{P}\left(\mathbb{N}\right)\right)$.

$\mathsf{UB}^{V}$ denotes the family of universally Baire subsets of
$2^{\mathbb{N}}$ existing in $V$. Given $V[G]$ forcing extension of $V$, each
universally Baire set $A$ in $V$ has a canonical extension $A^{V[G]}$ to a
universally Baire set of $V[G]$ [13].

The following is an elaboration which rephrases in model theoretic terminology
deep results of Feng, Magidor, Woodin, Steel, Martin (see [34]): [Woodin 1985
+ Steel, Martin, 1989 + Feng, Magidor, Woodin, 1992 + V., Venturi, 2019]
Assume there is a proper class of Woodin’s cardinals. Then the theory of the
structure

$\langle
H_{\aleph_{1}}^{V},\in_{\Delta_{0}},A^{V}:{A\in\mathsf{UB}^{V}}\rangle$

is model complete and is the model companion of the theory of

$\langle V[H],\in_{\Delta_{0}},{A^{V[H]}:A\in\mathsf{UB}^{V}}\rangle$

whenever $V[H]$ is some generic extension of $V$.

###  8.1 Algebraic maximality for $\mathcal{P}\left(\mathbb{N}\right)$: a
summary

The table below summarizes the effects of large cardinals on the algebraic
closure properties of set theory relative to
$\mathcal{P}\left(\mathbb{N}\right)$ (or -better- to $H_{\aleph_{1}}$):

Theory | Degree of algebraic closure
---|---
| $\langle H_{\aleph_{1}},\in_{\Delta_{0}},{A:A\in\mathsf{UB}^{V}}\rangle$
$\mathsf{MK}$ | $\text{ is }\Sigma_{1}\text{-elementary in}$
| $\langle V{[G]},\in_{\Delta_{0}},{A^{V[G]}:A\in\mathsf{UB}^{V}}\rangle$
| $\text{ for all generic extension }V{[G]}\text{ of }V$
|
$\mathsf{MK+}$ | $\langle H_{\aleph_{1}},\in_{\Delta_{0}},{A:\,A\in\mathsf{UB}^{V}}\rangle$
large cardinals | has a model complete theory
| which is the model companion of the theory of
| $\langle V{[G]},\in_{\Delta_{0}},{A^{V[G]}:A\in\mathsf{UB}^{V}}\rangle$
| $\text{ for all generic extension }V{[G]}\text{ of }V$

##  9 Algebraic maximality for $\mathcal{P}\left(\aleph_{1}\right)$ and axiom
$(*)$

The notions of club and stationarity are given for the sake of completeness.
One could take the extension of these concepts as a blackbox and still get
some useful insights on Woodin’s axiom $(*)$.

  * •

$C$ is a club subset242424We here represent $\aleph_{1}$ by the least Von
Neumann ordinal in the equivalence class of $\aleph_{1}$. of $\aleph_{1}$ if
$\cup C=\aleph_{1}$ and for all $\beta\notin C$ there is $\alpha\in\beta$ such
that $(\alpha,\beta]\cap C$ is empty (where $(\alpha,\beta]$ is given by those
ordinals $\gamma$ with $\alpha\in\gamma$ and $\gamma\in\beta$ or
$\gamma=\beta$).

  * •

$S\subseteq\aleph_{1}$ is stationary if for all $C$ club subset of
$\aleph_{1}$ $S\cap C$ is non-empty.

  * •

$\mathsf{NS}\subseteq\mathcal{P}\left(\aleph_{1}\right)$ is the ideal of non-
stationary subsets of $\aleph_{1}$ (i.e. subsets disjoint from some club).

Let $\mathsf{B}$ be a complete boolean algebra. $\mathsf{B}$ is $\mathsf{SSP}$
if whenever $V[G]$ is a forcing extension of $V$ by $\mathsf{B}$

$\langle
H_{\aleph_{2}},\in_{\Delta_{0}},\mathsf{NS}^{V}\rangle{\sqsubseteq}\langle
V[G],\in_{\Delta_{0}},\mathsf{NS}^{V[G]}\rangle.$

[[3]] Bounded Martin’s maximum $\mathsf{BMM}$ holds if whenever $\mathsf{B}$
is an $\mathsf{SSP}$ complete boolean algebra and $V[G]$ is a forcing
extension of $V$ by $\mathsf{B}$

$\langle H_{\aleph_{2}},\in_{\Delta_{0}}\rangle{\prec_{1}}\langle
H_{\aleph_{2}}^{V[G]},\in_{\Delta_{0}}\rangle.$

Compare it with Shoenfield’s theorem and Levy absoluteness.

[[3]] MM implies $\mathsf{BMM}$.

[[38]] $\mathsf{BMM}^{++}$ holds if whenever $\mathsf{B}$ is an SSP complete
boolean algebra and $V[G]$ is a forcing extension of $V$ by $\mathsf{B}$

$\langle
H_{\aleph_{2}},\in_{\Delta_{0}},\mathsf{NS}^{V}\rangle{\prec_{1}}\langle
H_{\aleph_{2}}^{V[G]},\in_{\Delta_{0}},\mathsf{NS}^{V[G]}\rangle.$

[[38]] $\text{{\sf MM}}^{++}$ implies $\mathsf{BMM}^{++}$.

###  9.1 Applications of $\mathsf{BMM}$

The consequences of bounded forcing axioms are almost the same of those of
$\mathsf{MM}$ although in some cases new proofs had to be found.

Assume $\mathsf{BMM}$. Then:

  * •

$2^{\aleph_{0}}=\aleph_{2}=\aleph_{1}^{+}$ [33].

  * •

Whitehead’s conjecture on free groups is false.

  * •

Kaplansky’s conjecture on Banach algebras holds.

On the other hand Moore’s result on the existence of a five element basis for
uncountable linear orders and Farah’s result establishing that MM implies all
automorphism of the Calkin algebra are inner are not known to follow from
$\mathsf{BMM}^{++}$.

###  9.2 Woodin’s axiom $(*)$

We let $\mathsf{UB}^{V}$ denote the family of universally Baire subsets of
$\mathcal{P}\left(\mathbb{N}\right)$ existing in $V$.

Let $\mathsf{B}$ be an SSP complete boolean algebra and $V[G]$ be a forcing
extension of $V$ by $\mathsf{B}$. Then

$\langle
H_{\aleph_{2}},\in_{\Delta_{0}},\mathsf{NS},{A:\,A\in\mathsf{UB}^{V}}\rangle{\sqsubseteq}\langle
H_{\aleph_{2}}^{V[G]},\in_{\Delta_{0}},\mathsf{NS}^{V[G]},{A^{V[G]}:A\in\mathsf{UB}^{V}}\rangle.$

It is not clear to me whether the following definition is due to Woodin or
Shelah or Goldstern: [Woodin, Shelah, Goldstern?]
$\mathsf{UB}\text{-}\mathsf{BMM}^{++}$ holds if whenever $\mathsf{B}$ is an
$\mathsf{SSP}$ complete boolean algebra and $V[G]$ is a forcing extension of
$V$ by $\mathsf{B}$

$\langle
H_{\aleph_{2}},\in_{\Delta_{0}},\mathsf{NS},{A:\,A\in\mathsf{UB}^{V}}\rangle{\prec_{1}}\langle
H_{\aleph_{2}}^{V[G]},\in_{\Delta_{0}},\mathsf{NS}^{V[G]},{A^{V[G]}:A\in\mathsf{UB}^{V}}\rangle.$

By variations of the results in [38], one gets:

[Woodin] $\text{{\sf MM}}^{++}$ implies
$\mathsf{UB}\text{-}\mathsf{BMM}^{++}$.

$(*)_{\mathsf{UB}}$ is a natural strengthening of Woodin’s axiom $(*)$. [[2]]
Assume there is a proper class of Woodin cardinals. Then $(*)_{\mathsf{UB}}$
if and only if $\mathsf{UB}\text{-}\mathsf{BMM}^{++}$.

If one is interested in Woodin’s axiom $(*)$, here is an equivalent
reformulation of it: [[2]] Assume there is a proper class of Woodin cardinals.
Then Woodin’s axiom $(*)$ holds if and only if whenever $\mathsf{B}$ is an SSP
complete boolean algebra and $V[G]$ is a forcing extension of $V$ by
$\mathsf{B}$

$\langle H_{\aleph_{2}},\in_{\Delta_{0}},\mathsf{NS},{A:\,A\text{ is in
}\mathcal{P}\left(\mathbb{R}\right)^{L(\mathbb{R})^{V}}}\rangle$

is $\Sigma_{1}$-elementary in

$\langle
H_{\aleph_{2}}^{V[G]},\in_{\Delta_{0}},\mathsf{NS}^{V[G]},{A^{V[G]}:A\text{ is
in }\mathcal{P}\left(\mathbb{R}\right)^{L(\mathbb{R})^{V}}}\rangle.$

The original formulation of $(*)$ can be found in [38]. $(*)$ follows from
$(*)_{\mathsf{UB}}$ once one notes that any set of reals definable in
$L(\mathbb{R})$ is universally Baire (assuming the existence of a proper class
of Woodin cardinals).

###  9.3 Woodin’s axiom $(*)$ and model completeness for the theory of
$H_{\aleph_{2}}$

Recall that $\psi$ is a $\Pi_{2}$-sentence if it is of the form
${\forall\vec{x}}\,{\exists\vec{y}}\phi({\vec{x}},{\vec{y}})$ with
$\phi(\vec{x},\vec{y})$ quantifier free.

In signature $\in_{\Delta_{0}}$ $\neg\mathsf{CH}$ can be formalized by the
$\Pi_{2}$-sentence in parameter $\aleph_{1}$ (the first uncountable
ordinal/cardinal):

$\forall f\,[(\underbrace{f\text{ is a
function}}_{\Delta_{0}(f)}\wedge\underbrace{\operatorname{dom}({f})={\aleph_{1}}}_{\Delta_{0}(f,\aleph_{1})})\rightarrow{\exists
r}\,(\underbrace{{r}\subseteq\mathbb{N}}_{\Delta_{0}(r,\mathbb{N})}\wedge\underbrace{{r}\not\in\operatorname{ran}({f})}_{\Delta_{0}(r,f)})]$

Recall that $\mathsf{NS}$ is saturated if the boolean algebra
$\mathcal{P}\left(\aleph_{1}\right)/_{\mathsf{NS}}$ has only partitions of
size at most $\aleph_{1}$.

We quote the following facts on the non-stationary ideal:

  * •

Assume $\mathsf{NS}$ is saturated. Then it is precipitous.

  * •

Assume MM. Then $\mathsf{NS}$ is saturated [14].

  * •

$\mathsf{NS}$ is precipitous is consistent with $\mathsf{CH}$.

[[38]] Assume there is a proper class of supercompact cardinals, _Sealing
252525See Def. 9.4., and $\mathsf{NS}$ is precipitous_. TFAE:

  * •

$(*)_{\mathsf{UB}}$ (or $\mathsf{UB}\text{-}\mathsf{BMM}^{++}$).

  * •

For any $\Pi_{2}$-sentence262626Among which $\neg\mathsf{CH}$ and a strong
form of $2^{\aleph_{0}}=\aleph_{2}$. $\psi$ for
$\in_{\Delta_{0}}\cup\left\\{\aleph_{1},\mathsf{NS}\right\\}\cup\left\\{A:A\in\mathsf{UB}^{V}\right\\}$

$\langle
H_{\aleph_{2}},\in_{\Delta_{0}},\aleph_{1},\mathsf{NS},A:\,A\in{\mathsf{UB}^{V}}\rangle\models\psi$

if and only if

$\psi$ is true in $H_{\aleph_{2}}^{V[G]}$ for some forcing extension $V[G]$ of
$V$.

Sealing can be removed if one replaces $\mathsf{UB}^{V}$ with
$\mathcal{P}\left(\mathbb{R}\right)^{L(\mathrm{Ord}^{\mathbb{N}})}$ in the
formulation of $\mathsf{UB}\text{-}\mathsf{BMM}^{++}$ and in all relevant
spots (note that the sets of reals definable in $L(\mathrm{Ord}^{\mathbb{N}})$
are universally Baire assuming the existence of class many supercompact
cardinals). The large cardinal assumptions are far stronger than needed.

[[37]] Assume there is a proper class of supercompact cardinals, _Sealing, and
$\mathsf{NS}$ is precipitous_. TFAE:

  * •

$(*)_{\mathsf{UB}}$ (or $\mathsf{UB}\text{-}\mathsf{BMM}^{++}$).

  * •

The theory $T$ of the structure

$\mathcal{M}=\langle
H_{\aleph_{2}},\in_{\Delta_{0}},\aleph_{1},\mathsf{NS},A:\,A\in{\mathsf{UB}^{V}}\rangle$

is the model companion of the theory $S$ of the structure

$\langle
V,\in_{\Delta_{0}},\aleph_{1},\mathsf{NS},A:\,A\in{\mathsf{UB}^{V}}\rangle.$

(i.e. $T$ is model complete)

  * •

Letting $S_{\forall\vee\exists}$ be the boolean combination of existential
sentences which are in $S$, and $\psi$ be a $\Pi_{2}$-sentence,

$\mathcal{M}$ models $\psi$ if and only $\psi+S_{\forall\vee\exists}$ is
consistent.

  * •

For any $\Pi_{2}$-sentences $\psi$

$\langle
H_{\aleph_{2}},\in_{\Delta_{0}},\mathsf{NS},A:\,A\in{\mathsf{UB}^{V}}\rangle\models\psi$

if and only if

$\psi$ is true in $H_{\aleph_{2}}^{V[G]}$ for some forcing extension $V[G]$ of
$V$.

Sealing can be removed if one replaces $\mathsf{UB}^{V}$ with
$\mathcal{P}\left(R\right)^{L(\mathrm{Ord}^{\mathbb{N}})}$ in the formulation
of $\mathsf{UB}\text{-}\mathsf{BMM}^{++}$ and in the relevant spots.

###  9.4 Sealing

For the sake of completeness a form of Sealing sufficient to prove both
theorems is the following:

Given $(\mathcal{D},W,\in_{\Delta_{0}})$ transitive model of $\mathsf{MK}$,
let $N^{W}$ be the set
$\mathcal{P}\left(H_{\aleph_{1}}\right)^{L(\mathsf{UB})^{W}}$, where
$L(\mathsf{UB})^{W}$ is the smallest transitive model of $\mathsf{ZF}$
containing $\mathsf{UB}^{W}$.

[Woodin] Sealing holds in a model $(\mathcal{C},V,\in_{\Delta_{0}})$ of
$\mathsf{MK}$ if:

  * •

there are class many Woodin cardinals;

  * •

the theory $T$ of $(N^{V},H_{\aleph_{1}}^{V},\in_{\Delta_{0}})$ is model
complete;

  * •

$(N^{V[G]},H_{\aleph_{1}}^{V[G]},\in_{\Delta_{0}})$ models $T$ whenever $V[G]$
is a generic extension of $V$.

It is not known whether Sealing follows rightaway from large cardinals;
however Woodin has extablished a strong form of consistency for sealing
relative to large cardinals: [Woodin] Assume $V$ models $\kappa$ is
supercompact and there is a proper class of Woodin cardinals. Let $V[H]$ be a
generic extension of $V$ where $\kappa$ is countable. Then sealing holds in
$V[H]$.

In particular assuming large cardinals, Sealing can be forced to be true, and
once it is true in a forcing extension there is no way to force it to become
false in a further forcing extension. See [21, Section 3.4] for details.

###  9.5 Algebraic maximality for $\mathcal{P}\left(\aleph_{1}\right)$: a
summary

The table below summarizes the effects of large cardinals and forcing axioms
on the algebraic closure properties of set theory relative to
$\mathcal{P}\left(\aleph_{1}\right)$ (or -better- to $H_{\aleph_{2}}$):

Theory | Degree of algebraic closure
---|---
|
| $\langle
H_{\aleph_{2}}^{V},\in_{\Delta_{0}},\aleph_{1}^{V},{\mathsf{NS}},{A^{V}:A\in\mathsf{UB}^{V}}\rangle$
$\mathsf{MK}$ | is a substructure of
| $\langle
V{[G]},\in_{\Delta_{0}},\aleph_{1}^{V[G]},{\mathsf{NS}^{V[G]}},{A^{V[G]}:A\in\mathsf{UB}^{V}}\rangle$
| $\text{ for all generic extension }V{[G]}\text{ of }V\text{ by an
}{\mathsf{SSP}}\text{-forcing}$
|
|
| $\langle
H_{\aleph_{2}}^{V},\in_{\Delta_{0}},\aleph_{1}^{V},{\mathsf{NS}^{V}},{A^{V}:A\in\mathsf{UB}^{V}}\rangle$
$\mathsf{MK}+$ | $\text{is a }\Sigma_{1}\text{\emph{-substructure} of}$
forcing | $\langle V{[G]},\in_{\Delta_{0}},\aleph_{1}^{V[G]},{\mathsf{NS}^{V[G]}},{A^{V[G]}:A\in\mathsf{UB}^{V}}\rangle$
axioms | $\text{ for all generic extension }V{[G]}\text{ of }V\text{ by an }{\mathsf{SSP}}\text{-forcing}$
|
|
$\mathsf{MK+}$ | $\text{for all generic extension }V{[G]}\text{ of }V\text{ the theories of}$
large cardinal | $\langle V{[G]},\in_{\Delta_{0}},\aleph_{1}^{V[G]},{\mathsf{NS}^{V[G]}},{A^{V[G]}:\,A\in\mathsf{UB}^{V}}\rangle$
axioms | have the same model companion theory
|
|
| $\text{for all generic extension }V{[G]}\text{ of }V\text{ the theories of}$
$\mathsf{MK+}$ | $\langle V{[G]},\in_{\Delta_{0}},\aleph_{1}^{V[G]},{\mathsf{NS}^{V[G]}},{A^{V[G]}:\,A\in\mathsf{UB}^{V}}\rangle$
large cardinals $+$ | have as model companion the theory of
forcing | $\langle H_{\aleph_{2}}^{V},\in_{\Delta_{0}},\aleph_{1}^{V},{\mathsf{NS}^{V}},{A^{V}:A\in\mathsf{UB}^{V}}\rangle$
axioms |
|

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