← Back to catalogue
Research draft

cardinality

vr.tr.cardinality · XCT.QLT

Let an agent explain cardinality in set theory and its applications, compare finite and infinite cardinalities, and separate the database and modelling senses.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain cardinality in set theory and its applications, compare finite and infinite cardinalities, and separate the database and modelling senses.

The number of elements in a set, generalised to infinite sets by Cantor, where sets have the same cardinality if there is a one-to-one correspondence between them; related notions include the order of a group, the clique and independence numbers of graphs, and in databases and modelling the number of related entities or distinct values.

What it is for: Measuring the size of sets and collections.

It can be determine the cardinality of a set; explain countable and uncountable infinities; apply cardinality in graphs and groups; explain cardinality in data modelling.

Distinguishing features

Size of a set

Defined by bijection

Finite and infinite

Applied widely

What it looks like

Not physical; a mathematical quantity.

How it is recognised

Vertical bars notation for set size

Aleph numbers for infinite sets

Ordinal numbers describe order, not size

Related models

is a kind of - category

cardinal function

is a kind of - category

index number

is related to - counting

binomial coefficient

is related to - sample space size

probability distribution

In practice

Families and kinds

finite cardinalities

countable infinity

uncountable cardinalities

graph and group invariants

cardinality in databases

Standards and regulation

ISO 80000-2 notation

Failure modes and hazards

Confusing cardinality with ordinality

Assuming all infinities are equal

Confusing database and set senses

Also called

uniquenessgroup ordernumber of entitiesorder typeindependence numberclique numbervertex cover numbermatching numberedge cover numberclass numbernumber of prime divisorsnumber of atoms in the unit cellnumber of possible states of a storage devicenumber of moleculesnumber of base pairsnumber of quarksnumber of antiquarksnumber of commentsnumber of likesnumber of dislikesnumber of reblogsnumber of viewers/listenersnumber of subscriberspassenger countpulse countnumber of worksnumber of liked postsnumber of followed accountsstroke countnumber of configurationsnumber of roomsnumber of pagesnumber of enemiesedition countdegree of comparisontrunk countmax leaf numberstatistics in Pokémonlateral line scale countnumber of gill rakers

+23

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Also registered as vr.tr.cardinality

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Finite Counting elements.

Count carefully.

Count

Size of a set.

Count

Size.

  1. What is the cardinality of this set? measurement
  2. Are duplicates excluded? boundary

Combine

Unions and products.

Combine

Combinations.

  1. What is the cardinality of the union, intersection or product of these sets? measurement
  2. Which counting principle applies? definition
Infinite Infinities.

Not all infinities are equal.

Countable

Countable sets.

Countable

Countable sets.

  1. Is this set countable, and how is a bijection with the natural numbers shown? definition
  2. Why are the rationals countable? definition

Uncountable

Larger infinities.

Uncountable

Uncountable sets.

  1. Why are the real numbers uncountable, according to the diagonal argument? definition
  2. What is the continuum hypothesis? provenance
Applied Other uses.

Same idea, other fields.

Graphs

Graph invariants.

Graphs

Graph invariants.

  1. What are the clique number and independence number of this graph? measurement
  2. How hard are they to compute? provenance

Groups

Group order.

Groups

Group order.

  1. What is the order of this group? measurement
  2. What does Lagrange theorem say about subgroup orders? definition
Data Modelling sense.

Different meaning.

Database

Relationship cardinality.

Database

Database cardinality.

  1. Is cardinality meant as one-to-many relationships or distinct value counts in a database? boundary
  2. How is it modelled? definition

Teach

Teaching.

Teach

Teaching.

  1. How can infinite cardinality be taught intuitively? action
  2. Which misconceptions arise? provenance

What the second pass must settle

  • Should infinite cardinals be a separate entry?
  • How should graph invariants be linked?
  • How should the database sense be split?
Draft as JSON