well-ordered set: meaning, definition, pronunciation and examples
Very LowTechnical / Academic
Quick answer
What does “well-ordered set” mean?
In mathematics, a set with a total order where every non-empty subset has a least element.
Audio
Pronunciation
Definition
Meaning and Definition
In mathematics, a set with a total order where every non-empty subset has a least element.
A precisely organized collection of items, usually where the ordering principle ensures that any sub-group has a defined 'first' or 'starting' element according to that order. This is a fundamental concept in set theory and order theory.
Dialectal Variation
British vs American Usage
Differences
No differences in meaning or usage. The term is identical in both mathematical communities.
Connotations
None beyond its strict mathematical definition.
Frequency
Identically rare and confined to technical mathematical discourse in both varieties.
Grammar
How to Use “well-ordered set” in a Sentence
The [set of natural numbers] is a well-ordered set.A well-ordered set under the [standard less-than relation].To show [X] is a well-ordered set, prove...Vocabulary
Collocations
Examples
Examples of “well-ordered set” in a Sentence
adjective
British English
- The well-ordered nature of the natural numbers is fundamental.
- She presented a proof relying on a well-ordered structure.
American English
- The well-ordered principle is equivalent to induction.
- We need a well-ordered index set for this construction.
Usage
Meaning in Context
Business
Never used.
Academic
Exclusively used in advanced mathematics, particularly set theory, logic, and foundations.
Everyday
Never used.
Technical
The primary domain of use. Refers to a specific structure with a rigorous definition.
Vocabulary
Synonyms of “well-ordered set”
Strong
Neutral
Weak
Vocabulary
Antonyms of “well-ordered set”
Watch out
Common Mistakes When Using “well-ordered set”
- Using it to mean simply 'neatly arranged'.
- Confusing it with a 'totally ordered set' (which lacks the condition on all subsets).
- Omitting the hyphen: 'well ordered set' is an acceptable variant, but 'well-ordered' is the standard adjectival form.
FAQ
Frequently Asked Questions
No, not under the standard order. While it is totally ordered, the subset of negative integers has no least element.
A totally ordered set requires that any two elements are comparable. A well-ordered set is a totally ordered set with the additional condition that every non-empty subset has a least (first) element.
Yes, absolutely. The set of natural numbers is the canonical example of an infinite well-ordered set.
It generalizes the principle of mathematical induction to sets of any size (transfinite induction) and is central to the study of ordinals and foundational set theory.
In mathematics, a set with a total order where every non-empty subset has a least element.
Well-ordered set is usually technical / academic in register.
Well-ordered set: in British English it is pronounced /ˌwel ɔːdəd ˈset/, and in American English it is pronounced /ˌwel ɔːrdərd ˈset/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: 'WELL-ORDERED' = In EVERY Little List, an Element is Clearly Evident Right at the Beginning. Every subset has a first element.
Conceptual Metaphor
ORDER IS LINEAR PROGRESSION (with a guaranteed starting point for any journey into the set).
Practice
Quiz
Which of the following is a defining property of a well-ordered set?