Concept
Uncountable set
Overview
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers.
Examples of uncountable sets include the set
R
{\displaystyle \mathbb {R} }
of all real numbers and the set of all subsets of the natural numbers.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
If every term of a collection can be hooked on to a number, and all the finite numbers are used once, and only once, in the process, then our collection must have just as many terms as there are finite numbers. This is the general method by which the numbers of infinite collections are defined.
Bertrand Russell, Mysticism and Logic and Other Essays
States what Uncountable set is.
(2) _Non-inductiveness._--The second property by which infinite numbers are distinguished from finite numbers is the property of non-inductiveness. This will be best explained by defining the positive property of inductiveness which characterises the finite numbers, and which is named after the method of proof known as "mathematical induction." Let us first consider what is meant by calling a property "hereditary"…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Uncountable set is.
When we think of mathematics, we have in our mind a science devoted to the exploration of number, quantity, geometry, and in modern times also including investigation into yet more abstract concepts of order, and into analogous types of purely logical relations. The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities.
Alfred North Whitehead, Science and the modern world
Close in meaning to Uncountable set; the term itself is not named.
Related ideas
- Fractal
- Georg Cantor
- Infinite set
- Zermelo–Fraenkel set theory
- Function (mathematics)
- Mathematics
- Real number
- Natural number
- Set theory
- Subset
- Bijection
- Cantor's diagonal argument
- Continuum hypothesis
- Countable set
- Cardinal number
- Cardinality
- Element of a set
- David Hilbert
- Axiom of choice
Place in the tradition
Part of First-order logic.