Reading Halmos' 'Naive Set Theory', I learn that these ZFC set axiom are used to recognise set that are set intuitively.
And the axiom of power, stated as 'For each set $X$, there exists a collection (set) of sets that contains among its elements all the subsets (again a set) of the given set.'
I am actually curious of whether there can be a 'collection' (not a set, it is a 'collection' of elements I have implicitly) 'inside' $X$ (not a subset, since is not a set).
This question may seem strange, one of the motivation is that in the power set $P(X)$, any element are subset of $X$ which is itself a set, I wonder if some 'collection' are already omitted. Another motivation is that any set to be chosen must be 'specific', that is, constucted under a sentence with set-builder notations (under axiom of specification).
I would like to formulate the statement and proof as follows, but fail even to state it.
Statement: Given a 'collection' of element (all $\in X$), it must form a set. (Prove or dispove)
Proof: ...
Problem I find in the statement: I already have in mind a 'collection' of objects, this seems like I can use axiom of specification to construct it as subset. But I really don't know what this set could be, if I really 'know' them, I am actually trying to index them (reducing unknown to known), this is already assuming they form a set (by considering range, which form a set by specification)
The whole problem lies on the definition of 'collection' (not a set). But in ZFC in the book, every objects to study are sets, so it does not really make sense.
Maybe another better way to formulate is if in a larger 'model', can there be defined another other object that is 'inside' $X$ (every element $\in X$) but not a set.
Sorry for being not specific, I will edit it later.