definition of countable set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of bijection.
Target Context
- The reader will have a definition of countable set.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*S\): \(\in \{\text{ the sets }\}\)
//
Conditions:
\(S \in \{\text{ the finite sets }\}\)
\(\lor\)
\(\exists f: \mathbb{N} \to S \in \{\text{ the bijections }\}\)
//
2: Note
The condition, "\(\exists f: \mathbb{N} \to S \in \{\text{ the bijections }\}\)", is equivalent to the condition, \(\exists f: \mathbb{N} \setminus \{0\} \to S \in \{\text{ the bijections }\}\), because there is the bijection, \(g: \mathbb{N} \to \mathbb{N} \setminus \{0\}, n \mapsto n + 1\), and \(f \circ g^{-1}\) or \(f \circ g\) is a bijection, by the proposition that any finite composition of bijections is a bijection, if the codomains of the constituent bijections equal the domains of the succeeding bijections.
If there is a surjection, \(f: \mathbb{N} \to S\), there is a bijection, \(f': \mathbb{N} \to S\), by the proposition that for any infinite set, if there is a surjection from the natural numbers set onto the set, there is a bijection from the natural numbers set onto the set, so, \(S\) is countable.
