A description/proof of that no 2 sets have each other as members
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 are no 2 sets such that each of them has the other as a member.
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 are no sets, \(S_1, S_2\), such that \(S_1 \in S_2\) and \(S_2 \in S_1\).
2: Proof
This proposition stems from the regularity axiom.
Suppose that \(S_1 \in S_2\) and \(S_2 \in S_1\). By the subset axiom, \(S'_1 := \{s \in S_1\vert s = S_2\} = \{S_2\}\) and \(S'_2 := \{s \in S_2\vert s = S_1\} = \{S_1\}\) would be sets. By the union axiom, \(S := S'_1 \cup S'_2 = \{S_1, S_2\}\) would be a set. \(S \cap S_1 \neq \emptyset\), because \(S_2 \in S\) and \(S_2 \in S_1\), and \(S \cap S_2 \neq \emptyset\), because \(S_1 \in S\) and \(S_1 \in S_2\), a contradiction, being against the regularity axiom.
