Why is the Power Set Operation Inherently Vague?

Question

It is a somewhat common view among mathematicians/philosophers (who have an opinion on the subject) that the power set operation is inherently vague. They go on to say that its inherent vagueness is the main reason that certain set-theoretic statements are absolutely undecidable. For example, Solomon Feferman, Nik Weaver , and Hartry Field explicitly hold this view.

I am seeking understanding for such a view. Namely, taking on board the meaningfulness of a set being "determinate" or "vague", I am asking what are the most compelling grounds for thinking:

(1): There exists an entirely definite set X such that $\mathcal P (X)$ is inherently vague.

The reason I am having trouble understanding such a view is because it seems that the only reason why (1) would be true is because one of the following two claims:

(2): There exists a set Y such that it is not definite whether Y is a subset of X.

(3): There exist elements of X such that it is not definite whether they form a set.

(2) seems to me to be false because the only way Y could fail to definitely be a subset of X, it seems, is that there exists some particular element $a \in Y$ such that it is inherently vague whether $a \in X$. But, this contradicts the fact that X is entirely determinat.

(3) seems to me to be false since it is an integral (i.e. not vague) part of our conception of the power set that any elements of a definite set X form a set. It is true that, say, ZFC cannot capture this line of thought since we only have the Axiom Schema of Separation saying definable subclasses of a set are sets, but the thought that any elements of a set form a set is an integral part of our conception of sets regardless. (It might be noticed that (3) is actually hard even to state since it appears to use second-order quantification over X, which when interpreted as quantification over subsets of X, says "There exists a subset of X, such that it might not REALLY be a subset". I don't know how persuasive saying that is in convincing someone that (1) is true.)

Any reasons/intuitions for why (2) or (3) are true, or why there is some other reason why (1) is true would be appreciated. Also, as a last question/reference request, are there any attempts at being more mathematically precise on having a theory of definiteness? Feferman very briefly sketches one in the above linked article having to do with intuitionist logic, but I can't find anyone that has tried to work on that.

Answer 1

I think one way of getting a grip on the vagueness is to explore multiple different power set operations and understand where they fall short and where they behave much like we would expect power set to behave. One straightforward one is to use $\mathbb{N}$ as the domain of discourse and look at the set of all finite subsets of a set $S$ (which I'll call $\mathcal{P}_f(S)$). This clearly doesn't satisfy all of the ZF axioms, but it does a remarkably good imitation of a power set (and, e.g., the set of all finite and cofinite subsets does an even better one). Once you feel like you have a handle on that and how it 'fits into' the rest of the axioms, you can consider the set of all constructible subsets of $S$ (for your favorite definition of constructible) and try to figure out where the problems slot in. In short, a lot of the vagueness of power set comes down to the notion of what constitutes a set in the first place, and particularly of how we can 'build' sets (and thus has very core connections to the axioms of specification/replacement/comprehension and to Russell's paradox).

Answer 2

Thats really a long comment that wouldn't fit the box:

I think of the real reason as (3):

Although our concept of set is definitely transitive, that is, if something is a set then every collection of elements of that thing must be a set, that is not true always, because, as you said, we only have separation axiom for first order formulas. If we had a general separation axiom, for example, we wouldn't need axiom of choice:

If $\mathcal{F}$ is family and we could extract subsets anyway we want, we could collect a subset of $\mathcal{P}(\mathcal{F})$ containing one element of each element of $\mathcal{F}$.

One instance of (1) happening is the problem of defining a choice function in ZF: The power set and separations are not powerful enough to determine if we can form a choice function (that is, specify a certain subset of the power set of the family)

Answer 3

Let's first look at (at least a 'semi-formal' rendition of) the Powerset Axiom (this from Suppes' book "Axiomatic Set Theory" found in the Dover edition on pg 47):

($\exists$B)($\forall$C)(C$\in$B iff C$\subseteq$A).

The question is:

"How should one interpret '($\forall$C)' in the axiom?"

For all intents and purposes there are two ways to interpret '($\forall$C)' in the axiom:

i) as '(all possible C)', or as ii) '(all C in a given domain of discourse)'--in this case a model of ZFC.

It should be noted that there is no way to define (i) in first-order logic. That leaves us with (ii), which restricts '($\forall$C)'to range over the domains of various and sundry models of ZFC (inner models, forcing extensions,....). Since (ii) does not allow one to speak of all possible subsets of A being collected into a single set, namely B, $\mathscr P$(A) is deemed to be...vague.