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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.

Taro
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  • For me, the vagueness is in the formation of the power set rather than the power set itself. I'm happy to say that the set of all subsets of $ \mathbb N $ exists - it's just not clear how one would go about forming it. – gamma Nov 07 '14 at 04:52
  • I definitely agree with you that's it's unclear to say the least how one would form it (especially if we are talking about some temporal process!), but from the perspective of absolute undecidability and set theoretic truth that outstrips ZFC, I feel like what I most care about is that, say, P(N) is perfectly definite. – Taro Nov 07 '14 at 04:56
  • The power set of any infinite set must be uncountable and therefore must contain uncountably many indefinite members in the sense that these members cannot be identified or defined. But words are not magic and our ability to identify something as definite is not a pre-requisite for its existence. Perhaps you are entertaining some constructivist / intuitionist ideas when you question if P( $ \mathbb N $ ) is definite. – gamma Nov 07 '14 at 05:03
  • I'm actually on the "realist" camp and tend to think that P(N) is perfectly fine. By posting this question I wanted to understand the other side. There are plenty of incredibly intelligent people who think that the power set operation is inherently vague and that this is the reason why, say, the Continuum Hypothesis can never have a truth value. (Also, the argument is not that it is vague whether or not P(N) exists, the argument is that the nature of P(N) is underdetermined by our conception. Like, when I say: "Consider a bald man, ..." he exists, but it is vague how many hairs he has.) – Taro Nov 07 '14 at 05:11
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    I think that your question is either too vague; or too broad for the context of this website. Either I don't get what you're trying to ask, or your question is more of a request for a philosophical discussion about set theory. In the former case, you should clarify; in the latter case, I'm not sure it is a good fit on this site. – Asaf Karagila Nov 07 '14 at 13:46
  • @AsafKaragila I tried to be explicit with the first two sentences of my last paragraph. As for whether its not a good fit, I'll defer to you since I am very much new to the site and you are very experienced. I feel like the existence of the "philosophy" tag is potentially misleading for new people on this site figuring out the social norms of what exactly is correct to ask... – Taro Nov 07 '14 at 13:54
  • There are philosophical questions that can be given a definite answer. I do agree it's a tricky tag, though. The issue here is the reasonable possibility of a definite answer of reasonable length. If the question hasn't got at least one of these, then it's not a good fit. Philosophy questions can be tricky like that, and have very long and broad answers written as books. – Asaf Karagila Nov 07 '14 at 14:04
  • Ok, my apologies. I'll refrain from posting questions like this in the future. It's sometimes hard to predict whether there can be "definite answers of reasonable length" before getting any of them, but I'll try to use that criterion. – Taro Nov 07 '14 at 14:13

3 Answers3

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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).

Steven Stadnicki
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  • Thanks. So it seems like you're endorsing (3) above. In response I said that it is part of our conception that ANY elements in X, which is by hypothesis perfectly definite, form a set (not just the constructible ones, or finite/cofinite ones) (Don't you agree that this is part of our conception?). What is your response? It can't be that it is vague what elements are in X since by hypothesis X was definite. – Taro Nov 08 '14 at 00:07
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    I think the catch in what you're saying is the 'ANY' - you've essentially built in a circularity to say that 'any subset of X is a subset of X', but we have no concrete means of forming an arbitrary collection of 'things in X', let alone the collection of all such collections. I think this is a good example, possibly the canonical example, of a case where instincts that have been groomed in the world of the finite fall apart on us in dealing with infinite sets. – Steven Stadnicki Nov 08 '14 at 00:18
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    (For instance, there are mathematicians who would actively disagree that there are any collections of the elements of X that aren't constructible!) – Steven Stadnicki Nov 08 '14 at 00:20
  • I agree its circular. Doesn't that means its a stand-still for both parties? The non-vague guy says ALL really means ALL and the vague guy says ALL doesn't mean ALL, and neither party really has an argument for their side. So I don't see this as a reason to think it is vague. I need a positive argument that ALL really can't mean ALL. The more substantial point is the following. Can't we do some informal mathematics here? Suppose for reductio that there existed some elements $x_\alpha$ that didn't form a set. Then, this would contradict the principle that ALL elements form a set. QED. – Taro Nov 08 '14 at 00:27
  • That's a bit tongue and cheek, but I just can't see where the vagueness comes in in saying ALL. If there were some counterexample, it would violate the principle. Therefore, there can't be a counterexample. Maybe this comes down to brute intuition clashes, but I just can't bring myself to sympathize with the other side at all. I think the crucial disagreement comes when you say "we have no concrete means of forming an arbitrary collection..." Since when does whether something exist depend on whether it can be formed! (says my platonist self) – Taro Nov 08 '14 at 00:30
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    Equally tongue-in-cheek, but pertinent how do you plan on identifying those elements that 'don't' form a set? – Steven Stadnicki Nov 08 '14 at 00:33
  • Why does this depend on whether I can identify them? If any such elements exist, they contradict the principle. So, to draw a contradiction, I only need their existence, not their name-ability or anything. Right? – Taro Nov 08 '14 at 00:36
  • You've just pushed the question up to what 'exists' means; there are sets of elements that exist in one model of set theory that don't exist in others, and the axioms of ZFC are - bluntly - not powerful enough to identify with that level of precision all of the possible sets. (Caveat: this is my perspective and doesn't necessarily reflect anyone else's) – Steven Stadnicki Nov 08 '14 at 00:41
  • I agree the axioms of ZFC are not up to the task, but note we are not considering the sets in the entire set-theoretic universe, we are only considering the elements that exist in some perfectly definite (by hypothesis) set X. So, this doesn't worry me. – Taro Nov 08 '14 at 00:44
  • But this discussion is getting too philosophical even for my own tastes, and I probably won't learn much more from discussions here, so I think I'll just "accept" your answer Steven since I got the most out of your discussion. – Taro Nov 08 '14 at 02:53
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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)

Jonas Gomes
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    I think its an uncontroversial fact that the power set operation, as viewed by ZFC, is vague. One need only look to forcing. My question was about whether our conception of the power set is vague, irrespective of formalizations. It seems to me that you might think that it is actually not vague, since our conception employs your "general separation axiom", which seems not to be vague. I'm basically looking for reasons why the "general separation axiom" is inherently vague. – Taro Nov 07 '14 at 04:23
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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.

Thomas Benjamin
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  • I did find this to be helpful, but when I situate this with respect to my formation of the question I'm not sure it has any force. What does this leave us with regards to (1)? Surely you don't think (1) is true for any X (e.g. you dont think the power set of the empty set is vague?), so what properties do we have to build in to X on your view to make (1) true? The reason I find your consideration unsatisfying is because we can just use bounded quantification for C (quantifying over subsets of A, which by hypothesis is determinate) and then say "nothing else is in B".(continued... – Taro Nov 07 '14 at 23:58
  • Do you think that quantification over subsets of a perfectly definite set is vague or do you think that the "nothing else is in B" part is vague? – Taro Nov 07 '14 at 23:59
  • @DavidBuiles: Consider the constructible universe L. $\mathscr P$($\omega$) restricted to L are just the constructible reals. Explicitly define a Cohen real c using L (if you can) and form L[c]. $\mathscr P$($\omega$) is now vague, at least in the sense you seem to use it in your question. – Thomas Benjamin Nov 08 '14 at 00:41
  • In this case, I agree that P($\omega$) is vague (i.e. differs in those two models) but I don't see how this is relevant to the question. In reality, P($\omega$) is misrepresented by both of these models, since it does not consist of merely constructible sets of natural numbers, but ALL sets of natural numbers. The crucial question is if this "ALL subsets" locution is vague when applied to a perfectly definite set. I'm not sure how your example is meant to illustrate this since I meant to include all non-constructible sets in "ALL" (see the discussion with Steven). – Taro Nov 08 '14 at 00:54
  • @DavidBuiles: But that's the point. Interpreting '$\forall$' as 'all possible' cannot be done in first-order logic. You need a modal ZFC with a revamped Powerset axiom (at least). That is why Powerset in first-order set theories is considered 'vague'. – Thomas Benjamin Nov 08 '14 at 01:07
  • Oh. Then we are talking past each other. I have always agreed that powerset in first-order set theories is vague, so you don't have to convince me there. But if they are only vaugue because of the limitations imposed in first-order theories and not otherwise, then the set theorist is open to saying, for example, CH is independent of ZFC due to the limitations of first order theories to capture the power set, but nonetheless CH has a definite truth value. My question was if powerset was inherently vague. – Taro Nov 08 '14 at 01:11
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    @DavidBuiles: Can you give me an example of a non-first-order powerset axiom with which you can explicitly define the notion of the set B of all possible subsets of a given set A? That would help me a lot, in more ways than one.... – Thomas Benjamin Nov 08 '14 at 01:16
  • I wish I could! If anyone could provide me with such an example I'd be ecstatic! Right now, I'm just trying to see the motivation of the logicians/philosophers who think that such a notion is just confused in some way or other. But as a side note, I'm not sure why we're using the adjective "possible" here. Extremely arbitrary nonconstructive subsets of $\omega$ are there, they aren't merely possibly there. – Taro Nov 08 '14 at 01:22
  • @DavidBuiles Why do you believe they absolutely ARE there? Especially in those circumstances where it's perfectly consistent that they're not there. I'm not saying that that's an unreasonable position to take - but it is a 'religious' position to take - there's no infallible concrete evidence for or against it. – Steven Stadnicki Nov 08 '14 at 01:57
  • I'm not a platonist who believes in the existence of mathematical objects independent of the physical world. I wouldn't call my position "religious" either. When we think about the naturals, I just think about a possible world where there are countably many particles. Every particle certainly exists, so to say that some collection of particles that is hard to name or describe doesn't actual exist in this world seems to me to be worse than religious. (I chose the naturals because they are concrete and many think their power set is vague. Of course this thought experiment is hard to generalize). – Taro Nov 08 '14 at 02:08
  • In this world my principle just says that all it means for a set of particles to exist is for each particle in that set to exist. So, I just find it uncontroversial, thinking about it this way, that the power set of the naturals is definite. – Taro Nov 08 '14 at 02:09
  • @DavidBuiles: 'Possible' and 'Necessary' are modal operators. The collection of 'All Possible' subsets of $\omega$, say, relative to a modal ZFC would be the union of the $\mathscr P$($\omega$) of all possible models of ZFC, which are the 'possible worlds'. – Thomas Benjamin Nov 08 '14 at 19:43
  • Oh, thanks for that! I'm familiar with modality in the context of metaphysical possible worlds, but applying it to models of ZFC looks like a nice idea! – Taro Nov 08 '14 at 19:51
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    @DavidBuiles: You might then want to take a look at the works of Hamkins and Lowe regarding the modal logic of forcing. Also of possible interest to you would be the works of Oystein Linnebo--he himself has developed a modal set theory--check out his homepage for preprints. – Thomas Benjamin Nov 09 '14 at 01:22
  • Thanks for the references! – Taro Nov 09 '14 at 01:23
  • @DavidB.: This is a late comment, but even if we grant the assumption that there are countably infinite particles, there will be collections of them that cannot be described, simply because every description is finite. This is precisely why the Henkin model of ZFC can be countable despite satisfying the power-set axiom. And there is certainly no way that physical entities can pinpoint them all. – user21820 Jan 21 '17 at 08:57