Ordinal number
An ordinal number is a type of number that is used to represent the position or rank of an object or a person in a list¹². For example, first, second, third, fourth, etc. are ordinal numbers. Most ordinal numbers end with “th” except when the final digit is one (first), two (second) or three (third)². In set theory, an ordinal number is also a generalization of ordinal numerals that can extend enumeration to infinite sets³.
What is the difference between cardinal and ordinal numbers?
The difference between cardinal and ordinal numbers is that cardinal numbers are used to count how many of something there are, while ordinal numbers are used to indicate the order or position of something in a list¹². For example, five, seven, eight, ten, etc. are cardinal numbers, while first, second, third, fourth, etc. are ordinal numbers. Cardinal numbers represent the quantity of an object or its magnitude, while ordinal numbers represent the location or position of an object in a given space¹³. Cardinal numbers are objective and quantitative, while ordinal numbers are subjective and qualitative¹³.
What are some examples of ordinal numbers in real life?
Some examples of ordinal numbers in real life are:
– The date of the month, such as January 1st, February 2nd, March 3rd, etc.
– The rank or position of a person or team in a competition, such as first place, second place, third place, etc.
– The order of events in a sequence, such as the first step, the second step, the third step, etc.
– The floor number of a building, such as the first floor, the second floor, the third floor, etc.
– The grade level of a student, such as first grade, second grade, third grade, etc.
How do you compare ordinal numbers?
To compare ordinal numbers, you can use different methods depending on the context and the format of the numbers. One method is to rewrite the ordinal numbers in a common form that makes them easily comparable, such as using numerals or words or symbols¹. For example, to compare 1st and 2nd, you can rewrite them as 1 and 2 or first and second or ω and ω + 1. Another method is to use a function that calculates the numeric values of the corresponding characters or symbols in each ordinal number²³. For example, to compare “first” and “second”, you can use the CompareOrdinal method in C# that returns -1 if “first” is less than “second”, 0 if they are equal, and 1 if “first” is greater than “second”. A third method is to use the properties of ordinal numbers and their operations, such as addition, multiplication, and exponentiation¹. For example, to compare ω + 2 and ω × 2, you can use the fact that ω × 2 = ω + ω > ω + 2.
Ordinals extend the natural numbers
Ordinal numbers extend the natural numbers by adding the order type of any well-ordered set. For example, the first infinite ordinal number is ω, which is the order type of the natural numbers. The next ordinal number is ω + 1, which is the order type of the natural numbers followed by a single element. Ordinal numbers can be added, multiplied, and exponentiated to form larger ordinals. The class of all ordinal numbers is well-ordered by the membership relation ∈.
What is a well ordered set?
A well-ordered set is a set that has a total order relation such that every non-empty subset of the set has a least element in this order¹². For example, the set of natural numbers with the usual order relation is a well-ordered set, because every non-empty subset of natural numbers has a smallest number. A well-ordered set has some properties that make it useful for mathematical reasoning, such as transfinite induction and ordinal numbers²³. A well-ordered set can also be characterized as a set that has a well-founded order, which means that there is no infinite descending sequence of elements in the order¹.
What is the difference between a well-ordered set and a totally ordered set?
The difference between a well-ordered set and a totally ordered set is that a well-ordered set has an additional property that every non-empty subset of the set has a least element in the order¹². A totally ordered set is a set that has a total order relation, which means that every pair of elements in the set is comparable, but it may not have a least element for every subset¹³. For example, the set of natural numbers with the usual order relation is a well-ordered set, but the set of real numbers with the usual order relation is a totally ordered set that is not well-ordered, because it has subsets that have no least element, such as the open interval (0, 1)²⁴. Every well-ordered set is a totally ordered set by definition, but not every totally ordered set is a well-ordered set²⁵.
What is Set theory?
Set theory is the branch of mathematics that deals with the properties of well-defined collections of objects, called sets, and their elements or members¹². Set theory can be used to study various kinds of mathematical objects, such as numbers, functions, relations, geometrical figures, etc., by representing them as sets or subsets of sets¹³. Set theory also provides the framework to develop a mathematical theory of infinity and to investigate the foundations of mathematics itself¹⁴. Set theory was initiated by Georg Cantor in the late 19th century and has since been developed by many mathematicians using different axiomatic systems and methods¹ .
What are some basic concepts and notation of set theory?
Some basic concepts and notation of set theory are:
– A set is a well-defined collection of objects, which can be denoted by listing its elements within curly braces { } or by using a defining rule¹². For example, A = {1, 2, 3} or B = {x : x is an even integer}.
– An element or a member of a set is an object that belongs to the set. The symbol ∈ is used to indicate that an element is in a set, and the symbol ∉ is used to indicate that an element is not in a set¹³. For example, 2 ∈ A and 4 ∉ A.
– A subset of a set is a set that contains only elements of the original set. The symbol ⊂ is used to indicate that a set is a subset of another set, and the symbol ⊄ is used to indicate that a set is not a subset of another set¹³. For example, {1, 2} ⊂ A and {1, 4} ⊄ A.
– Two sets are equal if they have exactly the same elements. The symbol = is used to indicate that two sets are equal¹³. For example, {1, 2, 3} = {3, 2, 1}.
– The empty set is the set that has no elements. It is denoted by ∅ or {}¹³. For example, ∅ ⊂ A and ∅ ≠ A.
– Some common sets of numbers are N (the natural numbers), Z (the integers), Q (the rational numbers), R (the real numbers), and C (the complex numbers)¹⁴. For example, N = {1, 2, 3, …}, Z = {…, -2, -1, 0, 1, 2, …}, Q = {m/n : m and n are integers and n ≠ 0}, R = {x : x can be written as a decimal expansion}, and C = {a + bi : a and b are real numbers and i = √(-1)}.
– Some operations on sets are union (∪), intersection (∩), difference (∖), and symmetric difference (Δ). The union of two sets is the set of all elements that are in either set. The intersection of two sets is the set of all elements that are in both sets. The difference of two sets is the set of all elements that are in the first set but not in the second set. The symmetric difference of two sets is the set of all elements that are in either set but not in both sets¹⁵. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∪ B = {1, 2, 3, 4}, A ∩ B = {2, 3}, A ∖ B = {1}, B ∖ A = {4}, and A Δ B = {1, 4}.
What are some applications of set theory?
Some applications of set theory are:
– Set theory is used in probability theory to model and calculate the likelihood of events and outcomes¹². For example, set operations such as union, intersection, and complement can be used to find the probability of the union or intersection of two events, or the probability of the complement of an event².
– Set theory is also used in logic and computer science to study and manipulate logical expressions and propositions¹³. For example, set operations such as union, intersection, and complement can be used to represent logical operations such as OR, AND, and NOT³.
– Set theory is also used in various branches of mathematics to define and analyze mathematical objects and structures¹⁴. For example, set theory can be used to define functions, relations, sequences, series, matrices, groups, rings, fields, etc., and to study their properties and operations⁴.
Notable People
Some famous set theorists are:
– Georg Cantor (1845-1918), who is considered the founder of set theory and who introduced the concepts of cardinality, ordinal numbers, transfinite arithmetic, and the continuum hypothesis¹².
– Richard Dedekind (1831-1916), who collaborated with Cantor and who developed the notion of Dedekind cuts and Dedekind-infinite sets¹².
– Bertrand Russell (1872-1970), who discovered the paradox that bears his name and who co-authored Principia Mathematica with Alfred North Whitehead, a monumental work that attempted to derive mathematics from logic and set theory¹³.
– Ernst Zermelo (1871-1953) and Abraham Fraenkel (1891-1965), who formulated the most widely accepted axiomatic system for set theory, known as Zermelo-Fraenkel set theory or ZF¹ .
– Kurt Gödel (1906-1978), who proved the completeness and incompleteness theorems for first-order logic and arithmetic, and who constructed a model of set theory in which the axiom of choice and the continuum hypothesis hold, known as Gödel’s constructible universe or L¹ .
– Paul Cohen (1934-2007), who developed the method of forcing and proved that the axiom of choice and the continuum hypothesis are independent of ZF, meaning that they can be neither proved nor disproved from the other axioms¹ .
– Ronald Jensen (born 1936), who proved that there is a smallest inner model of ZF containing all ordinals and all large cardinals, known as Jensen’s core model or K .
– W. Hugh Woodin (born 1955), who introduced many concepts and results in descriptive set theory, large cardinals, determinacy, inner models, and forcing axioms.
What are some paradoxes or problems in set theory?
Some paradoxes or problems in set theory are:
– Russell’s paradox, which shows that the naive set theory of Cantor and Dedekind leads to a contradiction by considering the set of all sets that are not members of themselves¹². This paradox motivated the development of axiomatic set theories that avoid such self-referential sets²³.
– Cantor’s paradox, which shows that there is no largest cardinal number by applying Cantor’s theorem to the set of all sets⁴ . This paradox reveals the existence of different levels of infinity and the need for a consistent way to compare them.
– Burali-Forti paradox, which shows that there is no largest ordinal number by applying the definition of ordinal numbers to the class of all ordinals⁴. This paradox exposes the distinction between sets and proper classes, and the limitations of naive set theory to deal with them .
– Richard’s paradox, which shows that there is no consistent way to define a set of all definable real numbers by using a diagonal argument similar to Cantor’s. This paradox raises questions about the notions of definability, computability, and language.
Q: What do you get when you cross a set with a mosquito?
A: A set that bites and a mosquito that stings.😂
Definition of an ordinal as an equivalence class
One way to define an ordinal as an equivalence class is as follows:
– An ordinal number is a type of number that represents the order type of a well-ordered set¹². A well-ordered set is a set that has a total order relation such that every non-empty subset of the set has a least element in this order²³.
– Two well-ordered sets are said to have the same order type if there exists a bijection between them that preserves the order relation. That is, if A and B are well-ordered sets with order relations ≤ and < respectively, then A and B have the same order type if there exists a function f : A → B such that for all x, y ∈ A, x ≤ y if and only if f(x) < f(y)²⁴.
– The order type of a well-ordered set is an equivalence class of well-ordered sets under the relation of having the same order type. That is, if A is a well-ordered set, then its order type is the set of all well-ordered sets that have the same order type as A² .
– An ordinal number can be identified with the order type of any well-ordered set that belongs to it. For example, the ordinal number 3 can be identified with the order type of the set {0, 1, 2} with the usual order relation² .
Von Neumann definition of ordinals
One way to explain the von Neumann definition of ordinals is as follows:
– The von Neumann definition of ordinals is a method of defining ordinals in set theory that uses the idea of representing an ordinal as the set of all smaller ordinals¹². For example, the ordinal number 3 can be represented as the set {0, 1, 2}, where 0, 1, and 2 are also ordinals defined as sets.
– The von Neumann definition of ordinals has some advantages over other definitions, such as being compatible with the axioms of Zermelo-Fraenkel set theory, being well-founded (meaning that there is no infinite descending sequence of ordinals), and being easy to compare and operate on²³.
– The von Neumann definition of ordinals can be formalized as follows: An ordinal is a transitive set that is well-ordered by the membership relation ∈. That is, an ordinal is a set α such that for all x and y in α, if x ∈ y then x ⊂ y (transitivity), and for all non-empty subsets S of α, there exists a least element z in S such that for all w in S, z ∈ w or z = w (well-ordering)²⁴.
– Using this definition, we can define some basic ordinals as follows: The empty set ∅ is an ordinal, called 0. For any ordinal α, the successor ordinal α + 1 is defined as α ∪ {α}. For any limit ordinal λ (an ordinal that is not 0 or a successor), λ is defined as the union of all smaller ordinals. For example, ω (the first infinite ordinal) is defined as the union of all finite ordinals² .
Transfinite sequence
One way to explain a transfinite sequence is as follows:
– A transfinite sequence is a generalization of a finite or infinite sequence that allows for indexing by any ordinal number¹². A sequence is a function that maps a set of indices (usually natural numbers) to a set of values (usually elements of some other set). A transfinite sequence is a function that maps an interval of ordinals [0, β) (where β is any ordinal) to a set of values¹³.
– A transfinite sequence can be used to construct or describe objects that depend on an ordinal number of steps or stages, such as ordinals themselves, cardinals, well-founded sets, recursive functions, etc²⁴. For example, using the von Neumann definition of ordinals, we can define a transfinite sequence f such that f(α) is the ordinal α for any ordinal α. Then f(0) = ∅, f(1) = {∅}, f(2) = {∅, {∅}}, f(ω) = {∅, {∅}, {∅, {∅}}, …}, etc² .
– A transfinite sequence can also be used to extend or generalize concepts or results from finite or infinite sequences to transfinite sequences. For example, we can define the limit of a transfinite sequence as the value that the sequence approaches as the index approaches a limit ordinal. We can also define convergence, divergence, monotonicity, boundedness, etc., for transfinite sequences using similar ideas as for finite or infinite sequences² .
Transfinite induction
One way to explain transfinite induction is as follows:
– Transfinite induction is an extension of mathematical induction to well-ordered sets, such as sets of ordinal numbers or cardinal numbers¹². Mathematical induction is a method of proving that a statement holds for all natural numbers by showing that it holds for 0 and that it holds for n + 1 whenever it holds for n²³. Transfinite induction generalizes this method to any well-ordered set, which is a set that has a total order relation such that every non-empty subset has a least element² .
– Transfinite induction can be used to prove statements about ordinals, cardinals, well-founded sets, recursive functions, and other objects that depend on an ordinal number of steps or stages² . For example, transfinite induction can be used to prove that every well-ordered set is order-isomorphic to a unique ordinal number² .
– Transfinite induction requires proving three cases: a base case, a successor case, and a limit case¹². The base case shows that the statement holds for the least element of the well-ordered set, usually 0. The successor case shows that the statement holds for any successor ordinal (an ordinal that has an immediate predecessor) whenever it holds for its predecessor. The limit case shows that the statement holds for any limit ordinal (an ordinal that has no immediate predecessor) whenever it holds for all smaller ordinals² .
Transfinite recursion
One way to explain transfinite recursion is as follows:
– Transfinite recursion is a generalization of finite or infinite recursion that allows for defining a function or an object by using an ordinal number of steps or stages¹². Recursion is a method of defining a function or an object by using a simpler or smaller version of itself. For example, we can define the factorial function n! by using recursion as follows: 0! = 1 and (n + 1)! = (n + 1) * n! for any natural number n²³. Transfinite recursion extends this method to any well-ordered set, which is a set that has a total order relation such that every non-empty subset has a least element² .
– Transfinite recursion can be used to construct or describe functions or objects that depend on an ordinal number of steps or stages, such as ordinals themselves, cardinals, well-founded sets, recursive functions, etc²⁴. For example, using the von Neumann definition of ordinals, we can define a function f such that f(α) is the ordinal α for any ordinal α by using transfinite recursion as follows: f(0) = ∅ and f(α + 1) = f(α) ∪ {f(α)} for any ordinal α and f(λ) = ∪{f(β) : β < λ} for any limit ordinal λ² .
– Transfinite recursion requires specifying three cases: a base case, a successor case, and a limit case¹². The base case defines the value of the function or the object for the least element of the well-ordered set, usually 0. The successor case defines the value of the function or the object for any successor ordinal (an ordinal that has an immediate predecessor) by using the value for its predecessor. The limit case defines the value of the function or the object for any limit ordinal (an ordinal that has no immediate predecessor) by using the values for all smaller ordinals² .
Successor and limit ordinals
One way to explain successor and limit ordinals is as follows:
– Successor and limit ordinals are two types of ordinal numbers, which are numbers that represent the order type of a well-ordered set¹². A well-ordered set is a set that has a total order relation such that every non-empty subset of the set has a least element in this order²³.
– A successor ordinal is an ordinal number that has an immediate predecessor, that is, another ordinal number that is smaller than it and that there is no ordinal number between them¹². For example, 1, 2, 3, ω + 1, ω + 2, ω + 3, etc., are successor ordinals. A successor ordinal can be obtained by adding 1 to any ordinal number² .
– A limit ordinal is an ordinal number that has no immediate predecessor, that is, for any ordinal number that is smaller than it, there is another ordinal number that is also smaller than it but larger than the first one¹². For example, ω, ω + ω, ω * ω, ω ^ ω, etc., are limit ordinals. A limit ordinal can be obtained by taking the supremum (the least upper bound) of any non-empty set of ordinals that has no greatest element² .
– Every ordinal number is either zero (the smallest ordinal), or a successor ordinal, or a limit ordinal¹². Successor and limit ordinals have different properties and behaviors in terms of arithmetic, topology, cardinality, etc² .
Indexing classes of ordinals
Indexing classes of ordinals is a way of identifying the α-th member of a class of ordinals, i.e. one can count them¹². A class of ordinals is closed and unbounded if its indexing function is continuous and never stops¹³⁴. This means that for any ordinal β, there is an ordinal α in the class such that α > β¹.
What are some examples of closed and unbounded classes of ordinals?
Some examples of closed and unbounded classes of ordinals are:
– The set of all countable limit ordinals, which is a club set with respect to the first uncountable ordinal ω1¹.
– The set of all limit ordinals less than a given uncountable initial ordinal κ¹.
– The range of any normal function, i.e. a function that is order-preserving and continuous²³.
– Any proper class of ordinals, which is unbounded in the class of all ordinals¹.
Closed unbounded sets and classes
Closed unbounded sets and classes are subsets or classes of ordinals that are closed and unbounded with respect to some limit ordinal. They are also called club sets, which is a contraction of “closed and unbounded”. They have many applications in set theory and logic, such as proving the existence of large cardinals and studying stationary sets.
Arithmetic of ordinals
Arithmetic of ordinals is the study of the three usual operations on ordinal numbers: addition, multiplication, and exponentiation¹². These operations can be defined either by constructing well-ordered sets that represent the result of the operation or by using transfinite recursion¹². Ordinal arithmetic has some properties that are different from the usual arithmetic of natural numbers, such as non-commutativity and non-cancellation¹². Ordinal arithmetic also has a standardized way of writing ordinals called the Cantor normal form¹².
Initial ordinal of a cardinal
The initial ordinal of a cardinal is the smallest ordinal that has that cardinal as its cardinality¹². Cardinality is the measure of the size of a set, ignoring the order of its elements¹². Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial¹². For example, the initial ordinal of the cardinal ℵ0 (aleph-null) is ω, the first infinite ordinal². The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal¹².
What is the axiom of choice?
The axiom of choice is a principle of set theory that states that given any collection of non-empty sets, it is possible to form a new set by arbitrarily choosing one element from each set¹²³. For example, if we have a collection of pairs of shoes, the axiom of choice allows us to pick out the left shoe from each pair to obtain a set of shoes¹. The axiom of choice was formulated by Ernst Zermelo in 1904 in order to prove the well-ordering theorem, which states that every set can be given an order relation such that every subset has a first element¹²³. The axiom of choice is equivalent to many other statements in mathematics, such as Zorn’s lemma and the existence of a basis for every vector space²³. However, the axiom of choice is also independent of the other axioms of set theory, meaning that it can be neither proved nor disproved from them²³. The axiom of choice has many useful applications, but also some paradoxical consequences, such as the Banach-Tarski paradox²³.
How can one prove the equivalence of the axiom of choice and Zorn’s lemma?
The equivalence of the axiom of choice and Zorn’s lemma can be proved by showing that each one implies the other¹²³⁴. Here is a sketch of the proof:
– To show that the axiom of choice implies Zorn’s lemma, assume that we have a partially ordered set P that satisfies the hypothesis of Zorn’s lemma, i.e. every chain in P has an upper bound in P. We want to show that P has a maximal element. To do this, we use the axiom of choice to construct a choice function f on the collection of all non-empty subsets of P. Then we use this function to define a transfinite sequence of elements of P by transfinite recursion. The idea is to start with any element p0 in P and then choose an element pα+1 that is strictly greater than pα using f whenever possible. If we reach a stage where this is not possible, then we have found a maximal element and we are done. Otherwise, we continue the process by taking limits at limit ordinals. We can show that this process must stop after at most |P| steps, where |P| is the cardinality of P, and that the last element of the sequence is maximal in P.
– To show that Zorn’s lemma implies the axiom of choice, assume that we have a collection A of non-empty sets. We want to show that there exists a choice function f on A. To do this, we consider the partially ordered set P of all partial choice functions on A, i.e. functions g with domain a subset of A such that g(X) is an element of X for every X in the domain of g. We order P by inclusion, i.e. g ≤ h if and only if g is a subfunction of h. We can show that P satisfies the hypothesis of Zorn’s lemma, i.e. every chain in P has an upper bound in P. This follows from the fact that the union of a chain of functions is again a function. By applying Zorn’s lemma, we conclude that P has a maximal element f. We claim that f is a choice function on A. To see this, suppose for contradiction that there exists a set X in A such that X is not in the domain of f. Then we can extend f to a larger function g by defining g(X) to be any element of X and g(Y) = f(Y) for all other Y in A. But this contradicts the maximality of f in P. Therefore, f must be defined on all of A and hence it is a choice function on A.