M. Foreman – Handbook of Set Theory (2010)

Автор: M. Foreman
Название книги: Handbook of Set Theory
Формат: PDF
Жанр: Математика
Страницы: 2212
Качество: Изначально компьютерное, E-book

It is fair to date the study of the foundation of mathematics back to the
ancient Greeks. The urge to understand and systematize the mathematics of
the time led Euclid to postulate axioms in an early attempt to put geometry
on a firm footing. With roots in the Elements, the distinctive methodology
of mathematics has become proof. Inevitably two questions arise: What are
proofs? and What assumptions are proofs based on?
The first question, traditionally an internal question of the field of logic,
was also wrestled with in antiquity. Aristotle gave his famous syllogistic systems,
and the Stoics had a nascent propositional logic. This study continued
with fits and starts, through Boethius, the Arabs and the medieval logicians
in Paris and London. The early germs of logic emerged in the context of
philosophy and theology.
The development of analytic geometry, as exemplified by Descartes, illustrated
one of the difficulties inherent in founding mathematics. It is classically
phrased as the question of how one reconciles the arithmetic with the geometric.
Are numbers one type of thing and geometric objects another? What are
the relationships between these two types of objects? How can they interact?
Discovery of new types of mathematical objects, such as imaginary numbers
and, much later, formal objects such as free groups and formal power series
make the problem of finding a common playing field for all of mathematics
importunate.
Several pressures made foundational issues urgent in the 19th century.
The development of alternative geometries created doubts about the view
that mathematical truth is part of an absolute all-encompassing logic and
caused it to evolve towards one in which mathematical propositions follow
logically from assumptions that may vary with context.
Mathematical advances involving the understanding of the relationship
between the completeness of the real line and the existence of solutions to
equations led inevitably to anxieties about the role of infinity in mathematics.
These too had antecedents in ancient history. The Greeks were well aware
of the scientific importance of the problems of the infinite which were put
forth, not only in the paradoxes of Zeno, but in the work of Eudoxus Archimedes and others. Venerable concerns about resolving infinitely divisible
lines into individual points and what is now called “Archimedes’ Axiom”
were recapitulated in 19th century mathematics.
In response, various “constructions” of the real numbers were given, such
as those using Cauchy sequences and Dedekind cuts, as a way of understanding
the relationship between discrete entities, such as the integers or
the rationals and the continuum. Even simple operations, such as addition
of arbitrary real numbers began to be understood as infinitary operations,
defined by some kind of limiting process. The notion of function was liberalized
beyond those that can be written in closed form. Sequences and series
became routine tools for solving equations.
The situation was made acute when Cantor, working on natural problems
involving trigonometric series, discovered the existence of different magnitudes
of infinity. The widespread use of inherently infinitary techniques,
such as the use of the Baire Category Theorem to prove the existence of important
objects, became deeply embedded in standard mathematics, making
it impossible to simply reject infinity as part of mathematics.
In parallel 19th century developments, through the work of Boole and others,
logic became once again a mathematical study. Boole’s algebraization of
logic made it grist for mathematical analysis and led to a clear understanding
of propositional logic. Dually, logicians such as Frege viewed mathematics as
a special case of logic. Indeed a very loose interpretation of the work of Frege
is that it is an attempt to base mathematics on a broad notion of logic that
subsumed all mathematical objects.
With Russell’s paradox and the failure of Frege’s program, a distinction
began to be made between logic and mathematics. Logic began to be viewed
as a formal epistemological mechanism for exploring mathematical truth,
barren of mathematical content and in need of assumptions or axioms to
make it a useful tool.
Progress in the 19th and 20th centuries led to the understanding of logics
involving quantifiers as opposed to propositional logic and to distinctions such
as those between first and second-order logic. With the semantics developed
by Tarski and the compactness and completeness theorems of G¨odel, firstorder
logic has become widely accepted as a well-understood, unproblematic
answer to the question What is a proof?
The desirable properties of first-order logic include:
• Proofs and propositions are easily and uncontroversially recognizable.
• There is an appealing semantics that gives a clear understanding of the
relationship between a mathematical structure and the formal propositions
that hold in it.
• It gives a satisfactory model of what mathematicians actually do: the
“rigorous” proofs given by humans seem to correspond exactly to the “formal” proofs of first-order logic. Indeed formal proofs seem to provide
a normative ideal towards which controversial mathematical claims
are driven as part of their verification process.
While there are pockets of resistance to first-order logic, such as constructivism
and intuitionism on the one hand and other alternatives such as
second-order logic on the other, these seem to have been swept aside, if simply
for no other reason than their comparative lack of mathematical fruitfulness.
To summarize, a well-accepted conventional view of foundations of mathematics
has evolved that can be caricatured as follows:
Mathematical Investigation = First-Order Logic + Assumptions
This formulation has the advantage that it segregates the difficulties with the
foundations of mathematics into discussions about the underlying assumptions
rather than into issues about the nature of reasoning.
So what are the appropriate assumptions for mathematics? It would be
very desirable to find assumptions that:
1. involve a simple primitive notion that is easy to understand and can be
used to “build” or develop all standard mathematical objects,
2. are evident,
3. are complete in that they settle all mathematical questions,
4. can be easily recognized as part of a recursive schema.
Unfortunately G¨odel’s incompleteness theorems make item 3 impossible. Any
recursive consistent collection A of mathematical assumptions that are strong
enough to encompass the simple arithmetic of the natural numbers will be
incomplete; in other words there will be mathematical propositions P that
cannot be settled on the basis of A. This inherent limitation is what has
made the foundations of mathematics a lively and controversial subject.
Item 2 is also difficult to satisfy. To the extent that we understand mathematics,
it is a difficult and complex business. The Euclidean example of
a collection of axioms that are easily stated and whose content is simple to
appreciate is likely to be misleading. Instead of simple, distinctly conceived
and obvious axioms, the project seems more analogous to specifying a complicated
operating system in machine language. The underlying primitive
notions used to develop standard mathematical objects are combined in very
complicated ways. The axioms describe the operations necessary for doing
this and the test of the axioms becomes how well they code higher level objects
as manipulated in ordinary mathematical language so that the results
agree with educated mathematicians’ sense of correctness.
Having been forced to give up 3 and perhaps 2, one is apparently left with
the alternatives:
2. Find assumptions that are in accord with the intuitions of mathematicians
well versed in the appropriate subject matter.
3. Find assumptions that describe mathematics to as large an extent as is
possible.
With regard to item 1, there are several choices that could work for the
primitive notion for developing mathematics, such as categories or functions.
With no a priori reason for choosing one over another, the standard choice
of sets (or set membership) as the basic notion is largely pragmatic. Taking
sets as the primitive, one can easily do the traditional constructions that
“build” or “code” the usual mathematical entities: the empty set, the natural
numbers, the integers, the rationals, the reals, C, Rn, manifolds, function
spaces—all of the common objects of mathematical study.
In the first half of the 20th century a standard set of assumptions evolved,
the axiom system called the Zermelo-Fraenkel axioms with the Axiom of
Choice (ZFC). It is pragmatic in spirit; it posits sufficient mathematical
strength to allow the development of standard mathematics, while explicitly
rejecting the type of objects held responsible for the various paradoxes,
such as Russell’s.
While ZFC is adequate for most of mathematics, there are many mathematical
questions that it does not settle. Most prominent among them is
the first problem on Hilbert’s celebrated list of problems given at the 1900
International Congress of Mathematicians, the Continuum Hypothesis.
In the jargon of logic, a question that cannot be settled in a theory T is
said to be independent of T. Thus, to give a mundane example, the property
of being Abelian is independent of the axioms for group theory. It is routine
for normal axiomatizations that serve to synopsize an abstract concept internal
to mathematics to have independent statements, but more troubling
for axiom systems intended to give a definitive description of mathematics
itself. However, independence phenomena are now known to arise from many
directions; in essentially every area of mathematics with significant infinitary
content there are natural examples of statements independent of ZFC.
This conundrum is at the center of most of the chapters in this Handbook.
Its investigation has left the province of abstract philosophy or logic and has
become a primarily mathematical project. The intent of the Handbook is
to provide graduate students and researchers access to much of the recent
progress on this project. The chapters range from expositions of relatively
well-known material in its mature form to the first complete published proofs
of important results. The introduction to the Handbook gives a thorough
historical background to set theory and summaries of each chapter, so the
comments here will be brief and general.
The chapters can be very roughly sorted into four types. The first type
consists of chapters with theorems demonstrating the independence of mathematical
statements. Understanding and proving theorems of this type require
a thorough understanding of the mathematics surrounding the source of the
problem in question, reducing the ambient mathematical constructions to
combinatorial statements about sets, and finally using some method (primarily
forcing) to show that the combinatorial statements are independent.
A second type of chapter involves delineating the edges of the independence
phenomenon, giving proofs in ZFC of statements that on first sight would
be suspected of being independent. Proofs of this kind are often extremely
subtle and surprising; very similar statements are independent and it is hard
to detect the underlying difference.
The last two types of chapters are motivated by the desire to settle these
independent statements by adding assumptions to ZFC, such as large cardinal
axioms. Proposers of augmentations to ZFC carry the burden of marshaling
sufficient evidence to convince informed practitioners of the reasonableness,
and perhaps truth, of the new assumptions as descriptions of the mathematical
universe. (Proposals for axiom systems intended to replace ZFC carry
additional heavier burdens and appear in other venues than the Handbook.)
One natural way that this burden is discharged is by determining what
the supplementary axioms say; in other words by investigating the consequences
of new axioms. This is a strictly mathematical venture. The theory
is assumed and theorems are proved in the ordinary mathematical manner.
Having an extensive development of the consequences of a proposed axiom
allows researchers to see the overall picture it paints of the set-theoretic universe,
to explore analogies and disanalogies with conventional axioms, and
judge its relative coherence with our understanding of that universe. Examples
of this include chapters that posit the assumption that the Axiom of
Determinacy holds in a model of Zermelo-Fraenkel set theory that contains
all of the real numbers and proceed to prove deep and difficult results about
the structure of definable sets of reals.
Were there an obvious and compelling unique path of axioms that supplement
ZFC and settle important independent problems, it is likely that the
last type of chapter would be superfluous. However, historically this is not the
case. Competing axioms systems have been posited, sometimes with obvious
connections, sometimes appearing to have nothing to do with each other.
Thus it becomes important to compare and contrast the competing proposals.
The Handbook includes expositions of some stunningly surprising
results showing that one axiom system actually implies an apparently unrelated
axiom system. By far the most famous example of this are the proofs
of determinacy axioms from large cardinal assumptions.
Many axioms or independent propositions are not related by implication,
but rather by relative consistency results, a crucial idea for the bulk of the
Handbook. A remarkable meta-phenomenon has emerged. There appears
to be a central spine of axioms to which all independent propositions are
comparable in consistency strength. This spine is delineated by large cardinal
axioms. There are no known counterexamples to this behavior.
Thus a project initiated to understand the relationships between disparate
axiom systems seems to have resulted in an understanding of most known
natural axioms as somehow variations on a common theme—at least as far
as consistency strength is concerned. This type of unifying deep structure is
taken as strong evidence that the axioms proposed reflect some underlying
reality and is often cited as a primary reason for accepting the existence of
large cardinals.
The methodology for settling the independent statements, such as the
Continuum Hypothesis, by looking for evidence is far from the usual deductive
paradigm for mathematics and goes against the rational grain of much
philosophical discussion of mathematics. This has directed the attention of
some members of the philosophical community towards set theory and has
been grist for many discussions and message boards. However interpreted,
the investigation itself is entirely mathematical and many of the most skilled
practitioners work entirely as mathematicians, unconcerned about any philosophical
anxieties their work produces.
Thus set theory finds itself at the confluence of the foundations of mathematics,
internal mathematical motivations and philosophical speculation. Its
explosive growth in scope and mathematical sophistication is testimony to
its intellectual health and vitality.
The Handbook project has some serious defects, and does not claim to be
a remotely complete survey of set theory; the work of Shelah is not covered to
the appropriate extent given its importance and influence and the enormous
development of classical descriptive set theory in the last fifteen years is
nearly neglected. While the editors regret this, we are consoled that those
two topics, at least, are well documented elsewhere. Other parts of set theory
are not so lucky and we apologize.
We the editors would like to thank all of the authors for their labors. They
have taken months or years out of their lives to contribute to this project.
We would especially like to thank the referees, who are the unsung heroes
of the story, having silently devoted untold hours to carefully reading the
manuscripts simply for the benefit of the subject.