A description/proof of that there is no set that contains all sets
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set.
Target Context
- The reader will have a description and a proof of the proposition that there is no set that that contains all the sets.
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: Description
There is no set that contains all the sets.
2: Proof
Let us suppose that there was a set that contains all the sets, \(C = \{S\vert S \text{ is any set}\}\). By the subset axiom, \(S' = \{s \in C\vert s \notin s\}\) would be a set. \(S' \in C\). If \(S' \in S'\), \(S' \notin S'\), and if \(S' \notin S'\), \(S' \in S'\), a contradiction.
