A description/proof of that part of set is subset if there is formula that determines each element of set to be in or out of part
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 any part of any set is a subset if there is a formula that determines each element of the set to be in or out of the part.
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: Note
The motivation of this proposition is that as the subset axiom in the ZFC set theory requires that the subset is defined with a legitimate formula, the axiom does not guarantees that a part of a set is a set (subset) unless a legitimate formula can be presented, even if each element is ambiguously determined whether it is in or out of the part.
2: Description
For any set, \(S\), and any formula, \(\phi\), such that for each \(p \in S\), exclusively \(\phi (p, 0)\) or \(\phi (p, 1)\), the part of \(S\), \(S_1:= \{p \in S\vert \phi (p, 1)\}\), is a set.
3: Proof
It is the subset axiom itself.
