- #1
dmuthuk
- 41
- 1
Hey everyone,
I am currently trying to learn a bit of set theory from Halmos' book "Naive Set Theory" since I have recently been concerned with the general notion of existence in various fields of mathematics.
Now, I am reading the "axiom of the power set" and I do find it a little troubling because I think that it might be possible to derive the existence of the power set from the axiom of specification (forming subsets), axiom of unions, axiom of pairing, and axiom of extentionality. So, I know have definitely gone wrong somewhere. Can someone help me find a flaw in the following argument:
Let E be a set. Each subset x of E exists by the axiom of specification. Then, each singleton set {x} exists by the axiom of pairing. By the axiom of unions, the union of this collection of singleton sets, each containing a subset of E as its only element, exists as a set. So, this is our desired power set. [QED]
Aside from this, I was wondering if anyone knows how to interpret the primitive relations of "equality" and "set membership" between any two sets. Can we think of these as "relations" on the set of all sets (which does not exist technically)?
Thanks,
Dilip
I am currently trying to learn a bit of set theory from Halmos' book "Naive Set Theory" since I have recently been concerned with the general notion of existence in various fields of mathematics.
Now, I am reading the "axiom of the power set" and I do find it a little troubling because I think that it might be possible to derive the existence of the power set from the axiom of specification (forming subsets), axiom of unions, axiom of pairing, and axiom of extentionality. So, I know have definitely gone wrong somewhere. Can someone help me find a flaw in the following argument:
Let E be a set. Each subset x of E exists by the axiom of specification. Then, each singleton set {x} exists by the axiom of pairing. By the axiom of unions, the union of this collection of singleton sets, each containing a subset of E as its only element, exists as a set. So, this is our desired power set. [QED]
Aside from this, I was wondering if anyone knows how to interpret the primitive relations of "equality" and "set membership" between any two sets. Can we think of these as "relations" on the set of all sets (which does not exist technically)?
Thanks,
Dilip