A description of relation between power set axiom and subset axiom
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 of a relation between the power set axiom and the subset axiom, when the ZFC set theory dictates that any element of any set is a 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: Note
In this article, we suppose that the ZFC set theory dictates that any element of any set is a set, which is, in fact, not any necessity and in fact, there are some theories that do not dictate so. But at least, mainstream theories seem to be dictating so.
2: Description
"Does the power set axiom not imply the subset axiom?" may be a question someone may conceive. He or she means, "As the power set is a set and any element of the power set is a set, is any subset not automatically a set, without the subset axiom?".
Well, any subset should be indeed a set, without the subset axiom, but what exactly are the subsets? The power set axiom implies that if there is a subset, it will be a set, but is not saying what subsets there are.
For example, for the natural numbers set, \(N\), the power set axiom is saying just that the set denoted by \(Pow N\) exists and any element of \(Pow N\) is a set called "subset". \(N\) and \(\emptyset\) are guaranteed to be subsets of \(N\) only by \(N = \{n \in N\vert n = n\}\) and \(\emptyset = \{n \in N\vert \lnot n = n\}\), using the subset axiom; \(\{1, 2\}\) is guaranteed to be a subset of \(N\) only by \(\{1, 2\} = \{n \in N\vert n = 1 \lor n = 2\}\).
Then, does a subset begin to exist only after we have given a formula for the subset axiom, \(\phi\), to define \(\{n \in N\vert \phi (n)\}\)? . . . No, let us dismiss such a humans-centric thought. The power set contains all the subsets without our giving each subset a formula: although we know the existence of a subset only after giving a formula, it is not that our knowing the existence makes it exist: what we do not know the existence of can exist without any help from humans. In other words, the formula should exist for any subset to exist, but humans do not need to know the formula for the subset to exist.
When we define \(N' := \{n' \in Pow N\vert \phi (n')\}\), \(\phi (n')\) is the sufficient condition for \(n'\) to belong to \(N'\) but is not necessarily a sufficient condition for \(n'\) to be a specific subset (generally, there can be multiple n's that satisfy \(\phi (n')\)), which means that no formula is given for \(n'\) to be a specific subset, but nevertheless, \(n'\) is guaranteed to be a subset.
