What is Ordinal: Definition and 26 Discussions

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another.
Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. The basic idea of ordinal numbers is to generalize this process to possibly infinite collections and to provide a "label" for each step in the process. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.
An ordinal number is used to describe the order type of a well-ordered set (though this does not work for a well-ordered proper class). A well-ordered set is a set with a relation < such that:

(Trichotomy) For any elements x and y, exactly one of these statements is true:
x < y
y < x
x = y
(Transitivity) For any elements x, y, z, if x < y and y < z, then x < z.
(Well-foundedness) Every nonempty subset has a least element, that is, it has an element x such that there is no other element y in the subset where y < x.Two well-ordered sets have the same order type, if and only if there is a bijection from one set to the other that converts the relation in the first set, to the relation in the second set.
Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, which are useful for quantifying the number of objects in a collection. Although the distinction between ordinals and cardinals is not always apparent in finite sets (one can go from one to the other just by counting labels), different infinite ordinals can correspond to the same cardinal. Moreover, there may be sets which cannot be well ordered, and their cardinal numbers do not correspond to ordinal numbers. (For example, the existence of such sets follows from Zermelo-Fraenkel set theory with the negation of the axiom of choice.) Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations is commutative.
Ordinals were introduced by Georg Cantor in 1883 in order to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872—while studying the uniqueness of trigonometric series.

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  1. F

    I When can Ordinal Variables be treated as Interval Variables?

    Hello, Ordinal variables (see Likert scale) can be labelled using numbers and ranked by those numbers. However, the difference between category 2 and category 3 may not be exactly be the same as the difference between category 4 and 5. That said, I noticed that in social science ordinal...
  2. A

    I Limits of Two Ordinal Sequences

    Let ##\omega_1## be the first uncountable ordinal such that ##x## is an element of ##\omega_1## if and only if it is either a finite ordinal or there exists a bijection from ##x## onto ##\omega##. I want to define a matrix such that the matrix contains each element of ##\omega_1## only once. To...
  3. A

    I Enumerating a Large Ordinal

    The following assertion quoted from the paper below seems as though it couldn’t be true. It is the issue that I would like some help addressing please: “The restriction of ##g(A)## to ##A \cap \omega_1## ensures that ##B## remains countable for this particular ##T## sequence.” ... Define...
  4. A

    I A Sequence T based on the Rule of Three

    Introduction: Making a Sequence ##T## based on “The Rule of Three” The primary means of generating the sequence ##T## is through the use of a function ##f##. In general, function ##f## is going to be a function that takes as input a three-member sequence of ordinal numbers (an ordered triplet)...
  5. J

    A Fundamental sequences for the Veblen hierarchy of ordinals

    I will first summarize the construction of ordinal numbers and introduce the definition of the binary Veblen function and of the notion of fundamental sequence. Ordinal numbers start with natural numbers 0, 1, 2, 3, ... which are followed by ## \omega ## which represents the "simple" infinity...
  6. FallenApple

    A What to do ordinal response variable?

    So what if my response variable, y, is say a scale. For example, ranking, something like 1-10. How would I transform the response to make linear regression work?
  7. G

    I Solving Ordinal Arithmetic: X Countably Compact but Not Compact

    Problem Statement: Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact. Let μ be the first uncountable ordinal. The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former...
  8. G

    I Ordinal Arithmetic: Proving X is Countably Compact

    Problem Statement: Show that the set X of all ordinals less than the first uncountable ordinal is countably compact but not compact. Let μ be the first uncountable ordinal. The latter question is easy to show, but I stumbled upon a curiosity while attempting the former. In showing the former...
  9. F

    MHB Set theory ordinal proof

    Let $\beta$ be an ordinal. Prove that $A\cap \bigcup\beta=\bigcup\{A\cap X\mid X \in \beta\}$ I'm not sure on this. It looks a bit like union distributing over intersection. Please help.
  10. W

    I Lack of Fit in Ordinal Regression -- Analysis/Alternatives?

    Hi All, I ran a binary logistic of Y on three different numerical variables A,B,C respectively. I am having an issue of separation of variables with all of them, meaning that there are values Ao,Bo, Co for each of A,B,C (different values for each, of course) so that for ## A>Ao, B>Bo...
  11. W

    I Are there Issues with Separation of Values in Ordinal Logistic Regression

    Hi all , just curious if someone knows of any issues of Separation of Points in Ordinal 3-valued Logistic Regression. I think I have an idea of why there are issues with separation in binary Logistic -- the need for the S-curve to go to 0 quickly makes the Bo term go to infinity. Are there...
  12. zrek

    I What is before the first transfinite ordinal, omega?

    Omega is the first transinite ordinal in the set of 0, 1, 2, ..., , , , ... This set is well ordered, so is after therefore is before . What is before ? Options: 1. -1 This is confusing, since if is the first transfinite ordinal, then -1 should be the last finite ordinal, which simply...
  13. P

    What is the name for an object that cannot be added to?

    What is the name for an object that cannot be added to (for example, a circle can only have 360 degrees; any more degrees and it is no longer merely a circle)?
  14. C

    Prove that ordinal arithmetic is associative

    Homework Statement Let a, b, c be ordinals. Prove that a+(b+c)=(a+b)+c Homework EquationsThe Attempt at a Solution I looked at a set theory book by Jech and he says Prove by induction on c. Should I look at the case where its true for c+1[/B]
  15. C

    Proving countable ordinal embeds in R

    Homework Statement show that if q is any countable ordinal, then there is a countable set A ⊆ R (in fact we can require A ⊆ Q), so that (A, <) ∼= (q, ∈). The Attempt at a Solution since q is a countable ordinal this implies that it has a mapping to the naturals. to me this seems strong enough...
  16. G

    What is the biggest ordinal that exists metamathematically?

    What is the biggest ordinal that exists metamathematically??
  17. T

    Seeking advice on creating ratio scales from ordinal scale instruments

    I have a psychological testing instrument that produces an ordinal measure (0, 1, 2, 3, 4, 5). I want to change this to a ratio scale with range 0 to 5. The instrument is designed so the first 5 questions are very easy (Level 1), the next 5 questions are harder (Level 2), the next set is...
  18. V

    Best textbook for learning about ordinal numbers

    HI - what is the best textbook, website or another resource to learn about ordinal numbers (Alph -0, Alph-1, etc) and their properties? I don't even know if there is a specific subfield of set theory that they fall under. Thanks!
  19. J

    Ordinal Property of Subsets in Well-Ordered Sets

    A set x is well-ordered by < if every subset of x has a least element. Here < is assumed a linear ordering, meaning that all members of a set can be compared, unlike with partial ordering. A set x is transitive if it has property \forall y\;(y\in x\to y\subset x). A set \alpha is ordinal, if...
  20. M

    Can ZF and ZFC both have consistent models?

    Why is there no number class containing all the ordinal numbers?
  21. D

    Ordinal Numbers: Bridging English & Set Theory

    How are ordinal numbers in set theory/order theory related to ordinal numbers in English? There should somehow be a bit of relationship for them to share the same name.
  22. A

    Cardinality of the set of ordinal numbers

    Does anyone happen to know what the cardinality of the set of ordinal number (transfinite and otherwise) is? A simplified proof would also be much appreciated. Recently I have been very interested in transfinite numbers and the logically gorgeous proofs involved :D
  23. I

    Does Every Ordinal Have a Following Cardinal?

    (not assuming any kind of continuum hypothesis of course) does every cardinal have a following one, i.e a minimal cardinal that is strictly larger?edit: ops, the title should be "following cardinal"
  24. Z

    Ordinal and Nominal Data: What is the Difference?

    I am having trouble distinguishing ordinal and nominal data, I know that ordinal is when there is some order but when you have to decide the type of data it is for something like "are you currently taking this medication?" it is very confusing as it seems like binary nominal to me but there is...
  25. L

    Proof theoretic ordinal of zfc, and other formal systems

    what is the proof-theoretic strength (largest ordinal whose existence can be proved) for ZFC set theory?
  26. U

    Hypothesis test on ordinal data

    Can i use the same the techniques used for nominal data on the ordinal data. I can't seem to find help on hypothesis test for ordinal data. This question contains categories (customer satisfaction) and they are in order so the data must be ordinal but they haven't taught us how to conduct a test...