tag

#set theory

18 pages tagged

Maths › 03. Logic and Foundations ›

9.5 Applications

Advanced set theory, especially forcing, large cardinals, and descriptive set theory, has deep applications in analysis and topology, because many central objects in these fields are sets of real numbers or functions, and their behavior depends strongly on definability, cardinality, and structural properties of the continuum. The purpose of this section is to explain how set theoretic principles influence classical areas of mathematics, often by determining whether certain pathological objects...
logic set theory analysis topology applications May 2, 2026
Maths › 03. Logic and Foundations ›

9.4 Determinacy

Determinacy principles study infinite games between two players, and they assert that for certain classes of games one of the two players has a winning strategy. These principles are important in set theory because they give strong regularity properties for definable sets of reals and provide a deep alternative perspective to the axiom of choice. Infinite Games We begin with games of perfect information played by two players, usually called...
logic set theory determinacy descriptive set theory May 2, 2026
Maths › 03. Logic and Foundations ›

9.3 Descriptive Set Theory

Descriptive set theory studies definable subsets of spaces such as the real line, the Baire space, and the Cantor space, and its main goal is to understand how the complexity of a set is related to the way the set can be described. The subject begins with topological spaces that are rich enough to support analysis but structured enough to allow precise classification, and the central examples are spaces whose...
logic set theory descriptive set theory definability May 2, 2026
Maths › 03. Logic and Foundations ›

9.1 Forcing

Forcing is a method for constructing new models of set theory from old ones by adjoining new sets in a controlled way, and its main purpose is to show that certain statements cannot be decided from a given list of axioms, such as ZFC, provided those axioms are consistent. The guiding idea is that we begin with a model $M$ of set theory and a partially ordered set $\mathbb{P}$ whose...
logic set theory forcing independence May 2, 2026
Maths › 03. Logic and Foundations ›

9. Advanced Set Theory

Advanced set theory studies methods and principles that go beyond the basic axioms of ZF and ZFC, especially when ordinary axioms do not decide important mathematical questions. The chapter begins with forcing, which is a method for constructing new models of set theory from old ones, and it provides the main technique for proving independence results such as the independence of the continuum hypothesis. Large cardinals are then introduced as...
logic set theory forcing large cardinals descriptive set theory May 2, 2026
Maths › 03. Logic and Foundations ›

9.2 Large Cardinals

Large cardinal theory studies strong axioms asserting the existence of infinite cardinals with properties far beyond those provable in ZFC, and these axioms are important because they measure the strength of set theoretic principles and provide a systematic hierarchy of stronger and stronger extensions of ordinary set theory. The basic idea is that some cardinals are so large that the universe below them resembles the whole universe in certain structural...
logic set theory large cardinals foundations May 2, 2026
Maths › 03. Logic and Foundations ›

8.5 Independence Phenomena

An independence result shows that a given statement cannot be proved or refuted from a given system of axioms, assuming that the system itself is consistent. In set theory, many natural and important statements turn out to be independent of the standard axioms. The study of independence phenomena reveals that the axioms of ZF and ZFC do not determine a single complete picture of the universe of sets. Instead, different...
logic set theory independence forcing constructibility May 2, 2026
Maths › 03. Logic and Foundations ›

8.1 Axiom of Choice

The axiom of choice is an additional principle of set theory which says that, given any family of nonempty sets, it is possible to choose one element from each set in the family. The statement sounds simple, but it has deep consequences because it applies even when the family is infinite and no explicit rule for making the choices is given. In ordinary finite mathematics, choosing one element from each...
logic set theory axiom of choice ZFC May 2, 2026
Maths › 03. Logic and Foundations ›

8.2 Equivalent Formulations

The axiom of choice has many equivalent forms. Some of these forms speak about selecting elements from sets, some speak about ordering arbitrary sets, and some speak about finding maximal objects in partially ordered sets. Although these statements look different, they express the same underlying principle over the usual axioms of ZF set theory. The purpose of this section is to explain the most important equivalent formulations in detail. The...
logic set theory axiom of choice zorn lemma well ordering May 2, 2026
Maths › 03. Logic and Foundations ›

8.3 Constructible Universe

The constructible universe is a class of sets built in stages by a controlled process of definability. It is denoted by: $$ L $$ The guiding idea is that, at each stage, we do not take all subsets of the previous stage. Instead, we take only those subsets that can be defined over the previous stage by a first order formula using parameters from that stage. In this way, $L$...
logic set theory constructible universe inner models May 2, 2026
Maths › 03. Logic and Foundations ›

8.4 Consistency Results

A consistency result explains that one formal theory does not lead to contradiction, usually relative to the consistency of another theory. In set theory, such results are especially important because many central principles cannot be proved or refuted from the standard axioms alone. The typical form is a relative consistency statement. Instead of proving absolutely that a theory is consistent, one proves that if a base theory is consistent, then...
logic set theory consistency inner models constructibility May 2, 2026
Maths › 03. Logic and Foundations ›

8. Axiomatic Systems

Axiomatic set theory studies the principles that govern the universe of sets, and it asks which mathematical statements follow from the chosen axioms, which statements require additional assumptions, and which statements cannot be decided from the usual axioms alone. The chapter begins with the Axiom of Choice, which asserts that one may choose an element from each set in a family of nonempty sets, and this principle has many consequences...
logic set theory axioms choice independence May 2, 2026
Maths › 03. Logic and Foundations ›

7.5 ZF and ZFC

Set theory becomes a foundation for mathematics only after its basic principles are stated as axioms. Informal set language is useful, but it can lead to contradictions if every definable collection is allowed to be a set. The Zermelo Fraenkel axioms provide a controlled formal framework in which sets may be built, compared, and used without allowing unrestricted comprehension. Motivation A naive view says that every collection described by a...
logic set theory ZF ZFC axioms May 2, 2026
Maths › 03. Logic and Foundations ›

7.4 Arithmetic of Cardinals

Cardinal arithmetic studies how the sizes of sets behave under natural set theoretic operations. Addition corresponds to disjoint union, multiplication corresponds to Cartesian product, and exponentiation corresponds to sets of functions. For finite sets these operations agree with ordinary arithmetic, but for infinite sets they have behavior that is at first surprising and later becomes one of the central tools of set theory. Cardinal Numbers A cardinal number is a...
logic set theory cardinals cardinal arithmetic infinity May 2, 2026
Maths › 03. Logic and Foundations ›

7.2 Cardinality and Countability

Cardinality is the part of set theory that compares the sizes of sets. For finite sets this agrees with ordinary counting, but for infinite sets the idea becomes more subtle, because two infinite sets may have the same size even when one appears to sit inside the other as a proper subset. Size by Correspondence The basic idea is that two sets have the same size when their elements can...
logic set theory cardinality countability infinity May 2, 2026
Maths › 03. Logic and Foundations ›

7.3 Ordinals and Well Ordering

Ordinals are set theoretic objects used to measure order type. Natural numbers measure the size of finite sets, while ordinals measure the structure of well ordered sets. The main idea is that a well ordered set is not only a collection of elements, but a collection arranged so that every nonempty subset has a first element. Well Ordered Sets A linear order tells us how to compare any two elements....
logic set theory ordinals well ordering transfinite induction May 2, 2026
Maths › 03. Logic and Foundations ›

7.1 Sets, Relations, Functions

Set theory provides a common language for modern mathematics. Many mathematical objects can be described as sets with additional structure, and many mathematical constructions can be described using membership, subsets, relations, and functions. In this section we introduce these basic notions carefully, since later parts of logic and foundations use them constantly. Sets and Membership A set is a collection of objects, called its elements or members. If an object...
logic set theory sets relations functions May 2, 2026
Maths › 03. Logic and Foundations ›

7. Basic Set Theory

Set theory provides a formal language for speaking about collections, membership, functions, relations, size, and order, and it serves as one of the standard foundations in which ordinary mathematical objects can be represented. The chapter begins with sets, relations, and functions, where the basic operations of mathematics are described in terms of membership and ordered pairs, giving a common framework for algebra, analysis, topology, and logic. Cardinality and countability are...
logic set theory cardinality ordinals zfc May 2, 2026