Ordinals and well-orderings are fundamental concepts in theory and logic. They extend the idea of counting to infinite sets, allowing us to compare and measure infinite quantities. This topic is crucial for understanding the structure of infinite sets and their relationships.
In the theory of recursive functions, ordinals play a key role in measuring computational complexity. They provide a framework for classifying functions and sets based on their level of computability, forming hierarchies that help organize and analyze recursive processes.
Definition of ordinals
Ordinals are a generalization of the that extend the concept of counting to infinite sets
Ordinals capture the notion of order type, which is the abstract structure of a well-ordered set
In the theory of recursive functions, ordinals play a crucial role in measuring the complexity of computations and defining hierarchies of functions
Ordinals as order types
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An ordinal represents the order type of a well-ordered set, which is a set with a total order where every non-empty has a least element
Two have the same order type (ordinal) if there exists an order-preserving bijection between them
Ordinals abstract away the specific elements of a well-ordered set and focus on the essential structure of the ordering
Transfinite ordinals
Transfinite ordinals are ordinals that extend beyond the finite ordinals (natural numbers)
The smallest transfinite ordinal is denoted by ω and represents the order type of the natural numbers
Transfinite ordinals allow us to describe and compare the sizes of infinite sets in a precise manner
Successor and limit ordinals
Every ordinal has a successor, which is the smallest ordinal greater than it
For a finite ordinal n, its successor is n+1, while for a transfinite ordinal α, its successor is denoted by α+1
Limit ordinals are ordinals that are not successors of any other ordinal
Examples of limit ordinals include ω, ω+ω, and ω2
The class of ordinals is closed under the successor operation and taking limits of increasing sequences
Ordinal arithmetic
extends the usual arithmetic operations (addition, multiplication, exponentiation) to ordinals
Ordinal arithmetic satisfies certain properties that differ from usual arithmetic on natural numbers
Ordinal arithmetic is essential for understanding the structure and behavior of ordinals
Ordinal addition
Ordinal addition is defined by
For any ordinal α and finite ordinal n, α+n is defined as the n-th successor of α
For any ordinals α and β, α+β is defined as the limit of α+γ for all γ<β
Ordinal addition is associative but not commutative (1+ω=ω=ω+1)
Ordinal multiplication
Ordinal multiplication is also defined by transfinite recursion
For any ordinal α and finite ordinal n, α⋅n is defined as the sum of n copies of α
For any ordinals α and β, α⋅β is defined as the limit of α⋅γ for all γ<β
Ordinal multiplication is associative, left-distributive over addition, but not commutative (2⋅ω=ω=ω⋅2)
Ordinal exponentiation
is defined by transfinite recursion as well
For any ordinal α and finite ordinal n, αn is defined as the product of n copies of α
For any ordinals α and β, αβ is defined as the limit of αγ for all γ<β
Ordinal exponentiation is right-distributive over multiplication and satisfies the power law (αβ)γ=αβ⋅γ
Properties of ordinals
Ordinals possess several important properties that distinguish them from other number systems
These properties are crucial for understanding the behavior of ordinals and their role in the theory of recursive functions
The properties of ordinals form the foundation for many results and applications in set theory and logic
Well-ordering of ordinals
The class of ordinals is well-ordered under the usual order relation <
For any two distinct ordinals α and β, exactly one of α<β, α=β, or β<α holds
Every non-empty set of ordinals has a least element, which allows for proofs by
Trichotomy law for ordinals
The trichotomy law states that for any two ordinals α and β, exactly one of α<β, α=β, or β<α holds
This law is a consequence of the of ordinals and is stronger than the comparable property for real numbers
The trichotomy law is essential for comparing ordinals and proving statements about their relationships
Induction on ordinals
Transfinite induction is a powerful proof technique that extends the principle of mathematical induction to ordinals
To prove a statement P(α) for all ordinals α, it suffices to show:
P(0) holds
If P(β) holds for all β<α, then P(α) holds
Transfinite induction allows for proofs involving infinite sequences and limit stages, making it a fundamental tool in ordinal theory
Countable ordinals
are ordinals that have a countable number of predecessors
The class of countable ordinals is itself uncountable and plays a significant role in the theory of recursive functions
Countable ordinals provide a way to measure the complexity of certain recursive processes
Definition of countable ordinals
An ordinal α is countable if there exists a bijection between α and a subset of the natural numbers
Equivalently, an ordinal is countable if it is either finite or has the same as ω
The class of countable ordinals is denoted by ω1 and is the first uncountable ordinal
Church-Kleene ordinal
The , denoted by ω1CK, is the least non-recursive ordinal
An ordinal is recursive if it is the order type of a computable well-ordering of the natural numbers
The Church-Kleene ordinal represents the limit of and is a key concept in the theory of recursive functions
Recursive ordinals
An ordinal is recursive if it is the order type of a computable well-ordering of the natural numbers
Recursive ordinals are countable and form a proper subclass of the countable ordinals
The class of recursive ordinals is closed under ordinal arithmetic and provides a hierarchy of computable functions
Uncountable ordinals
are ordinals that have an uncountable number of predecessors
The study of uncountable ordinals involves deep questions in set theory and leads to the development of large cardinal axioms
Uncountable ordinals provide a way to measure the size and complexity of mathematical structures beyond the countable realm
Continuum hypothesis
The (CH) states that there is no cardinal number strictly between ℵ0 (the cardinality of the natural numbers) and 2ℵ0 (the cardinality of the real numbers)
CH is independent of the standard axioms of set theory (ZFC), meaning it can neither be proved nor disproved from these axioms
The status of CH has significant implications for the structure of uncountable ordinals and the properties of sets of real numbers
Alephs and beth numbers
(ℵα) and (ℶα) are two hierarchies of cardinal numbers used to describe the sizes of infinite sets
Alephs are defined by transfinite recursion: ℵ0=ω and ℵα+1 is the least cardinal greater than ℵα
Beth numbers are defined by transfinite recursion: ℶ0=ω and ℶα+1=2ℶα
The generalized continuum hypothesis (GCH) states that ℵα=ℶα for all ordinals α, extending CH to all infinite cardinals
Inaccessible cardinals
An uncountable cardinal κ is inaccessible if it is regular (not the sum of fewer than κ smaller cardinals) and strong limit (2λ<κ for all λ<κ)
The existence of cannot be proved in ZFC and requires stronger axioms
Inaccessible cardinals are used to construct models of set theory and study the consistency and independence of mathematical statements
Well-ordered sets
Well-ordered sets are sets equipped with a total order in which every non-empty subset has a least element
The study of well-ordered sets is closely related to the theory of ordinals, as every well-ordered set has a unique ordinal (its order type) associated with it
Well-ordered sets provide a foundation for the theory of ordinals and have applications in various areas of mathematics
Definition of well-ordered sets
A set X is well-ordered by a relation < if:
< is a total order on X (reflexive, antisymmetric, transitive, and total)
Every non-empty subset of X has a least element with respect to <
Well-ordered sets can be finite or infinite, and the natural numbers with the usual order form a well-ordered set
Isomorphism of well-ordered sets
Two well-ordered sets (X,<X) and (Y,<Y) are isomorphic (or order-isomorphic) if there exists a bijection f:X→Y such that for all a,b∈X, a<Xb if and only if f(a)<Yf(b)
Isomorphic well-ordered sets have the same order type (ordinal) and share the same structural properties
The isomorphism relation is an equivalence relation on the class of well-ordered sets
Existence of non-isomorphic well-orderings
For any infinite cardinal κ, there exist well-ordered sets of cardinality κ that are not isomorphic to each other
This result demonstrates the richness and diversity of well-orderings beyond the countable case
The existence of non-isomorphic well-orderings has implications for the structure of ordinals and the study of uncountable sets
Ordinals in set theory
Ordinals play a central role in set theory, providing a foundation for the study of well-ordered sets and the hierarchy of infinite sets
In set theory, ordinals are constructed as transitive sets that are well-ordered by the membership relation ∈
The study of ordinals in set theory leads to deep questions about the nature of infinity and the consistency of mathematical systems
Ordinals as transitive sets
A set X is transitive if every element of X is also a subset of X, i.e., for all a∈X and b∈a, we have b∈X
In set theory, ordinals are defined as transitive sets that are well-ordered by the membership relation ∈
This definition allows for a canonical representation of ordinals and provides a way to study their properties using the tools of set theory
Von Neumann ordinals
are a specific construction of ordinals as transitive sets, named after mathematician John von Neumann
The von Neumann ordinal corresponding to a well-ordered set (X,<) is defined as the set {{y∈X:y<x}:x∈X}
Von Neumann ordinals satisfy the property that every element of an ordinal is also a subset of it, i.e., α∈β implies α⊂β
Burali-Forti paradox
The , named after mathematician Cesare Burali-Forti, is a paradox in naive set theory that arises from considering the set of all ordinals
The paradox shows that the collection of all ordinals cannot be a set, as it would lead to a contradiction
The resolution of the Burali-Forti paradox in axiomatic set theory (such as ZFC) involves recognizing that the class of all ordinals is a proper class, not a set
Applications of ordinals
Ordinals have numerous applications in various areas of mathematics, including proof theory, computer science, and the study of formal systems
The use of ordinals allows for the precise measurement of complexity, the classification of structures, and the development of powerful proof techniques
Understanding the applications of ordinals is crucial for appreciating their significance and the role they play in the foundations of mathematics
Ordinal notations
Ordinal notations are systems for representing ordinals using symbols and rules for manipulating those symbols
Common ordinal notations include the Cantor normal form, the Veblen hierarchy, and the Bachmann-Howard ordinal
Ordinal notations provide a way to express and compare large ordinals, enabling the study of their properties and the development of proof-theoretic results
Ordinal analysis in proof theory
Ordinal analysis is a technique in proof theory that assigns ordinals to formal theories to measure their strength and consistency
The proof-theoretic ordinal of a theory is the least ordinal that cannot be proved to be well-ordered within the theory
Ordinal analysis has been used to establish the consistency and independence of various mathematical systems, such as subsystems of second-order arithmetic
Ordinals in computer science
Ordinals have applications in computer science, particularly in the study of termination and complexity of algorithms
Ordinal-indexed functions and data structures, such as the Hardy hierarchy and the fast-growing hierarchy, are used to analyze the growth rates of computable functions
Ordinals also appear in the study of infinite games, where they are used to define winning strategies and classify the complexity of game-theoretic problems
Key Terms to Review (29)
Alephs: Alephs are symbols used to denote the cardinalities of infinite sets in set theory, particularly in the context of comparing different sizes of infinity. They represent various types of infinity, with the smallest aleph, denoted as \( \aleph_0 \), representing the cardinality of countably infinite sets, such as the set of natural numbers. Alephs play a crucial role in understanding the hierarchy of infinities and how they can be well-ordered.
Beth numbers: Beth numbers are a sequence of cardinal numbers used to describe the sizes of infinite sets, beginning with the smallest infinite cardinal, denoted as \( \beth_0 \), which corresponds to the cardinality of the set of natural numbers. Each subsequent beth number is defined as the power set of the previous beth number, illustrating a hierarchy of infinities. This concept is important in understanding different sizes of infinity and their relationship with ordinals and well-orderings.
Burali-Forti Paradox: The Burali-Forti Paradox arises in set theory and concerns the existence of a 'largest ordinal.' It shows that if you assume that such an ordinal exists, you can construct a contradiction, revealing that the collection of all ordinals cannot be well-ordered. This paradox highlights significant issues in defining and understanding ordinals and well-orderings, particularly concerning the limits of our set-theoretic constructs.
Cantor's Theorem: Cantor's Theorem states that for any set, the power set of that set (the set of all its subsets) has a strictly greater cardinality than the set itself. This theorem reveals the existence of different sizes of infinity, establishing that not all infinities are equal and connecting deeply to the concepts of ordinals and well-orderings.
Cardinality: Cardinality refers to the measure of the 'number of elements' in a set, which helps categorize sets based on their size. It distinguishes between finite sets, which have a specific number of elements, and infinite sets, which can be countably or uncountably infinite. Understanding cardinality is essential in comparing different sets and understanding their structure, particularly in the context of ordinals and well-orderings.
Church-Kleene ordinal: The Church-Kleene ordinal is the smallest ordinal that cannot be recursively represented, serving as a pivotal concept in recursion theory and the study of computability. It arises from the exploration of hyperarithmetic sets and is key to understanding the limitations of recursive functions. This ordinal is often used to illustrate the boundary between computable and non-computable sets, providing insights into well-orderings and recursive pseudo-well-orderings.
Continuum hypothesis: The continuum hypothesis posits that there is no set whose cardinality is strictly between that of the integers and the real numbers. This concept has profound implications in set theory and the study of infinite sets, particularly regarding the nature of different sizes of infinity. It raises questions about the structure of the continuum and how various sizes of infinite sets relate to one another in the context of ordinals and well-orderings.
Countable Ordinals: Countable ordinals are well-ordered sets that can be put into a one-to-one correspondence with the natural numbers, meaning they can be 'counted' or indexed by them. These ordinals represent different 'sizes' or types of infinity and play a crucial role in understanding the structure of well-ordered sets and the hyperarithmetical hierarchy. They provide a framework for comparing different infinite processes, allowing us to study their properties and relations in a precise manner.
Georg Cantor: Georg Cantor was a mathematician known for founding set theory and introducing the concept of infinite numbers, particularly different sizes of infinity. His work laid the groundwork for understanding ordinals and well-orderings, which are essential in analyzing the structure of sets, especially in the context of recursive ordinals where well-ordered sets play a crucial role in defining recursive processes.
Inaccessible cardinals: Inaccessible cardinals are a special type of large cardinal that cannot be reached by any set of smaller cardinal numbers through standard set-theoretic operations. They are larger than all cardinals that can be formed from smaller cardinals, making them crucial in the study of set theory and the hierarchy of infinities. Inaccessible cardinals play a significant role in understanding the structure of the set-theoretic universe and in discussions about the foundations of mathematics.
Kurt Gödel: Kurt Gödel was an Austrian-American logician, mathematician, and philosopher, best known for his groundbreaking work on the incompleteness theorems. His contributions have had a profound impact on various fields, including the limitations of formal systems, computability, and set theory.
Limit ordinal: A limit ordinal is an ordinal number that is not zero and cannot be reached by adding 1 (or any finite number) to a smaller ordinal. Instead, it is defined as the least upper bound of all smaller ordinals. Limit ordinals are essential in understanding recursive ordinals and help establish a hierarchy within the broader framework of ordinals and well-orderings.
Natural Numbers: Natural numbers are the set of positive integers that start from 1 and extend indefinitely (1, 2, 3, ...). They play a fundamental role in various mathematical concepts, particularly in defining sequences, counting, and establishing the basis for recursion and inductive definitions, as well as in understanding well-ordering principles.
Ordinal arithmetic: Ordinal arithmetic refers to the operations of addition, multiplication, and exponentiation defined for ordinals, which are well-ordered sets that extend the concept of natural numbers. These operations do not always behave as they do with natural numbers; for instance, ordinal addition is not commutative, meaning that the order of the operands matters. Understanding ordinal arithmetic is essential for exploring recursive ordinals, establishing connections between ordinal notations, and developing recursive pseudo-well-orderings.
Ordinal exponentiation: Ordinal exponentiation is a mathematical operation that extends the concept of exponentiation to ordinal numbers, defining a way to raise one ordinal to the power of another. This operation is not as straightforward as exponentiation with natural numbers, as it incorporates the well-ordering of ordinals and can lead to results that differ from the intuitive properties seen in finite or even infinite cardinal exponentiation. Understanding this concept helps in grasping the nature of larger ordinals and their behaviors in various mathematical contexts.
Ordinal number: An ordinal number is a concept in mathematics that represents the position or order of elements within a well-ordered set. Unlike cardinal numbers, which indicate quantity, ordinal numbers specify a rank or sequence, such as first, second, and third. This property is essential when discussing well-orderings, as it allows for the arrangement of sets where every subset has a least element.
Recursive ordinals: Recursive ordinals are a specific type of ordinal numbers that can be defined or generated by recursive processes. These ordinals serve as important benchmarks in understanding the hierarchy of computable functions and the limits of algorithmic processes. Their significance extends to various concepts like the hyperarithmetical hierarchy, well-orderings, and ordinal notations, making them foundational in the study of mathematical logic and computation.
Set: A set is a well-defined collection of distinct objects, considered as an object in its own right. Sets are fundamental in mathematics and are used to describe relationships and structures, such as ordinals and well-orderings, which are crucial for understanding order types and hierarchies within mathematical theory.
Subset: A subset is a set formed from another set, containing some or all of the elements of that original set. The concept of subsets is crucial in understanding the relationships between sets, especially when discussing ordinal numbers and their well-orderings. Each subset can have its own properties and structures, which can help in analyzing ordered sets and their various elements.
Successor Ordinal: A successor ordinal is an ordinal number that comes immediately after another ordinal in the ordering of ordinals. It is formed by taking an existing ordinal and adding one to it, representing the next step in the well-ordered set of ordinals. Successor ordinals play a crucial role in understanding the hierarchy of ordinals and how they are structured, particularly when considering their properties in relation to limit ordinals and well-orderings.
Transfinite induction: Transfinite induction is a method of mathematical proof used to establish properties for all ordinals by extending the principle of mathematical induction to transfinite ordinals. It allows one to prove that a certain property holds for an ordinal number by assuming it holds for all smaller ordinals and then demonstrating that it also holds for the ordinal in question. This technique is vital in discussing inductive definitions, recursion, recursive ordinals, and the well-ordering of ordinals.
Transfinite Recursion: Transfinite recursion is a method for defining functions on ordinals, extending the concept of ordinary recursion to transfinite cases. This approach allows for the creation of functions that can take into account the entire hierarchy of ordinals, leading to definitions that are well-suited for dealing with infinite structures and processes. By using transfinite recursion, one can build complex functions in a systematic way, ensuring that the values at each stage depend on previous stages, making it essential for understanding advanced topics like recursive ordinals and the Church-Kleene ordinal.
Uncountable ordinals: Uncountable ordinals are a class of ordinals that cannot be put into a one-to-one correspondence with the natural numbers, meaning they are larger than any countable ordinal. They represent a type of well-ordered set that extends beyond the limits of countable sequences and are crucial for understanding the structure of ordinal numbers and their properties. These ordinals play a key role in set theory, especially in discussions around transfinite numbers and their applications in mathematical logic.
Von Neumann ordinals: Von Neumann ordinals are a type of ordinal number where each ordinal is identified with the set of all smaller ordinals. This means that for any ordinal $$eta$$, it can be represented as the set containing all ordinals less than $$eta$$. This construction forms a well-ordered set and is crucial in understanding the foundation of ordinal arithmetic and transfinite induction.
Well-ordered sets: A well-ordered set is a type of ordered set where every non-empty subset has a least element. This characteristic allows for a clear and systematic way to compare and organize elements within the set. Well-ordered sets are closely linked to ordinals, which are used to represent the order type of well-ordered sets, facilitating a deeper understanding of their properties and applications in various mathematical contexts.
Well-ordering: Well-ordering is a property of a set that states every non-empty subset has a least element under a given ordering. This concept is crucial in understanding the structure of ordinals, where every ordinal can be well-ordered, leading to important implications in recursion and hierarchies in mathematical logic.
Well-Ordering Theorem: The well-ordering theorem states that every non-empty set of ordinals can be well-ordered, meaning that every subset has a least element. This theorem plays a critical role in set theory and the study of ordinals, as it connects the concept of order types with the structure of sets, particularly in establishing the properties of ordinal numbers and their applications in various mathematical contexts.
α: In the context of ordinals and well-orderings, α typically represents an ordinal number, which is a way to describe the order type of well-ordered sets. Ordinals extend beyond natural numbers and can represent infinite quantities, allowing us to understand various levels of infinity and their properties. The concept of α plays a crucial role in set theory and mathematical logic, particularly in discussions about the hierarchy of infinite sets and the structure of well-ordered sets.
β: In the context of ordinals and well-orderings, β represents a specific ordinal number that serves as a limit ordinal. It signifies the point at which every smaller ordinal is included, and no greater ordinal can be formed by simply adding one to it. Understanding β is crucial in studying the properties of well-ordered sets and their cardinalities, particularly when discussing transfinite sequences.