well-ordering theorem: meaning, definition, pronunciation and examples
Very Low (C2)Formal, Technical
Quick answer
What does “well-ordering theorem” mean?
A mathematical theorem stating that every set can be well-ordered.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical theorem stating that every set can be well-ordered; i.e., it can be given a total order such that every non-empty subset has a least element.
In set theory, the statement that every set can be equipped with a well-order. It is equivalent to the Axiom of Choice. In philosophical and foundational discussions, it refers to the principle enabling the ordering of any collection, no matter how large or complex.
Dialectal Variation
British vs American Usage
Differences
No lexical or spelling differences. Usage is identical across mathematical communities.
Connotations
None beyond the technical mathematical meaning.
Frequency
Identically very rare outside specialist academic contexts.
Grammar
How to Use “well-ordering theorem” in a Sentence
The well-ordering theorem [VERB] that...By the well-ordering theorem, [CLAUSE]Vocabulary
Collocations
Examples
Examples of “well-ordering theorem” in a Sentence
adjective
British English
- a well-ordering theorem proof
American English
- a well-ordering theorem result
Usage
Meaning in Context
Business
Not used.
Academic
Used exclusively in advanced mathematics, set theory, and mathematical logic lectures, papers, and textbooks.
Everyday
Not used.
Technical
The primary context. Discussed in relation to the Axiom of Choice, Zermelo-Fraenkel set theory, and transfinite induction.
Vocabulary
Synonyms of “well-ordering theorem”
Neutral
Weak
Watch out
Common Mistakes When Using “well-ordering theorem”
- Confusing it with the Well-Ordering *Principle* for the natural numbers (which is a property, not an equivalent of the Axiom of Choice).
- Using it in non-mathematical contexts.
- Capitalising it inconsistently (usually not capitalised except at the start of a sentence).
FAQ
Frequently Asked Questions
They are logically equivalent within standard set theory (ZF). This means if you accept one as true, you must accept the other, but they are phrased as different statements.
No, it is an existence theorem. It proves that a well-order *exists* for any set, but it does not provide a constructive method for defining it.
The natural numbers {1, 2, 3, ...} with the usual 'less than' order are well-ordered. Every non-empty subset (like the even numbers) has a smallest element.
No. The usual '<' order on real numbers is not a well-order. For example, the open interval (0, 1) has no least element. The well-ordering theorem states a different, non-constructible well-order *exists* for the reals.
A mathematical theorem stating that every set can be well-ordered.
Well-ordering theorem is usually formal, technical in register.
Well-ordering theorem: in British English it is pronounced /ˌwel ˈɔː.dər.ɪŋ ˌθɪə.rəm/, and in American English it is pronounced /ˌwel ˈɔːr.dɚ.ɪŋ ˌθɪr.əm/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Imagine a librarian who can find the 'first' (smallest) book on *any* shelf, no matter how chaotically the shelf is stocked. The well-ordering theorem says such a librarian (a well-order) exists for *every possible collection* of books (sets).
Conceptual Metaphor
ORGANIZING THE UNORGANIZABLE: The theorem is a guarantee that even the most chaotic, infinite, or complex collection can be arranged in a single, neat, step-by-step line where every step has a clear 'next' item.
Practice
Quiz
The well-ordering theorem is primarily a concept from which field?