# Regarding my question: "Can only the constructible sets be proven to exist in $ZF$...."

I am concerned that the 'bad reviews' regarding my most recent question are more harassment than helpful. They seem to come from distinctions between $Def$ (or what I call $\mathscr P_{Def}$), $\mathscr P_{L}$, and the usual power set operation $\mathscr P$. The answers and comments given seem to suggest that I have somehow confused $\mathscr P_{Def}$ with $\mathscr P_{L}$ (although $\mathscr P_{Def}$ is frequently called--though possibly wrongly--the "Predicative Power Set Operation"). That the 'reviewers' have answered the questions in the manner they have (for though $ZF$ can possibly prove that there exist constructible sets, it cannot prove the existence (without extra axioms) the existence of non-constructible sets) makes it impossible to delete the question without possible penalty. And that to me makes the situation impossible to rectify (and give my 'critics' a means to continually downvote me). If I have made the 'gaffe' of confusing $\mathscr P_{Def}$ with $\mathscr P_{L}$ then they should kindly tell me and let me make the necessary corrections or, if not, they should produce answers more relevant to the issues I raise. How can I best deal with the situation? Thanks in advance.

• I really wish people would link to their concern on MO main rather than make others hunt it down. mathoverflow.net/questions/278720/…
– Todd Trimble Mod
Aug 24, 2017 at 16:30
• I guess this should be tagged (specific-question), see the tag-info. Aug 24, 2017 at 16:35
• (Martin: maybe, but in the first place I am skeptical that this question is on-topic here.) Thomas, you need to ask them mathematical questions directly at the question, as opposed to raising charges of "harassment" here, and putting "reviewers" in inverted commas and other forms of rudeness. You are certainly free to edit the question yourself, if you determine from discussion there that you've committed conceptual errors.
– Todd Trimble Mod
Aug 24, 2017 at 16:42
• Thomas, in my previous comment I interpreted "reviewers" as people like Asaf and Noah (since you refer to reviewers who answered the questions in the manner they have). In other words, the reviewers here seem to be people who have honestly tried to engage with the question and give helpful feedback. They interpreted the question as best they could and tried to answer the question thus interpreted (Noah's seems very helpful to me, as he wrote down a formula for a set that ZF cannot prove is constructible.) If he didn't answer the question you meant, nothing stops you from further clarification.
– Todd Trimble Mod
Aug 25, 2017 at 12:14
• So, one suggestion is to engage Noah below his answer, if you consider his answer not to be relevant to the issues you raise. Actually I think his answer might be worth fleshing out (either publicly on MO or in the privacy of your room), because what one even means by $0^\sharp$ is somewhat subtle, due to Tarski's result on undefinability of truth.
– Todd Trimble Mod
Aug 25, 2017 at 13:30
• @ToddTrimble: very helpful. Thanks. I will mention your observation in a comment to him (giving you due credit, of course). Aug 25, 2017 at 14:24
• Could you please clarify what do you mean by That the 'reviewers' have answered the questions in the manner they have (for though ZF can possibly prove that there exist constructible sets, it cannot prove the existence (without extra axioms) the existence of non-constructible sets) makes it impossible to delete the question without possible penalty.? Aug 26, 2017 at 11:56
• (@მამუკაჯიბლაძე As a point of actual "meta", the software prevents the OP from deleting his post because there are upvoted answers; this has nothing to do with the topic of his post of course. What I don't understand is what comes after that, where supposedly "the situation [is] impossible to rectify". But it may be best to let tempers cool for a while.)
– Todd Trimble Mod
Aug 26, 2017 at 13:32

I am one of the people who has downvoted this question (which I did after answering it, based on how the situation evolved from there, although to be honest I did suspect I would downvote from the beginning based on previous experience). I did this for a couple reasons. I didn't mention these in the main question, but I also didn't intend them to be secret or anything - I thought they would be clear from context.

Given this meta question, though, I'd like to say a bit about why I downvoted, since I think this will address the question you should be asking here, namely "What can I do better?"

First, the question is already somewhat unclear (and was more unclear to begin with), but more problematically attempts to clarify it haven't yielded a lot of success (and it's worth noting that both votes to close currently are as "unclear"). I'm still not entirely sure whether either Asaf's or my answer addresses the question you're trying to ask, although in my opinion they address the only reasonable question you could be asking here. Of course sometimes one starts with a somewhat unclear question and then finds it clarified over time, but I don't think that has happened in this case, at least not enough.

In particular, you write here:

They should produce answers more relevant to the issues I raise.

The answers we've produced are directly relevant to the only meaningful interpretation of your question I can come up with; I genuinely don't understand what is missing from them. (Note that all this is addressing your question (i); your question (ii), meanwhile, seems to be completely answered by the observation that $\mathcal{P}$ and $\mathcal{P}_{def}$ never coincide in any model of ZF, but there seems to be some confusion on this point.) If we're unable to come up with answers you think are relevant, then maybe you haven't succeeded in asking the question you want to.

Second, I strongly believe that you're putting the cart before the horse: basic confusions (e.g. the conflation between models and theories in the original formulation, confusions re: sets versus formulas defining sets, and questions like "can ZF prove some sets are constructible?"/"can [ZF] prove that there exist constructible sets" that have emerged in the comments) and how you've responded to them make me doubt whether, even if the question were phrased clearly, an answer can be given to you at the moment.

Even beyond that, the way you've engaged the answers and subsequent comments has left me in the dark about what you do and don't understand. Of main relevance is:

Do you understand that ZF can prove the existence of sets whose constructibility ZF cannot prove (e.g. ZF proves that $\mathcal{P}(\omega)$ exists, but cannot prove that $\mathcal{P}(\omega)$ is constructible)?

(If you do, this takes us back to my first point: how does this not answer your question?)

And the following arises from the conversation in the comments, although it's not really directly relevant to your question:

Do you understand that, and why, we can never have $\mathcal{P}_{def}=\mathcal{P}$ in any model of ZF?

Basically, from your responses to my and Asaf's answers and the various comments, I don't think you're currently in a place where this question can be satisfactorily answered; moreover, I don't have a good enough understanding of what you do understand to help lead you in a good direction.

Finally, there's the issue of your general engagement with this question (and other questions); this subsumes the previous point. What you've done here and elsewhere is try to draw your question into a larger philosophical web of ideas; the problem is that this only introduces more confusions, especially when the mathematical content of the question is already unclear. That broader web of ideas is something you can only engage after developing a solid command of the basics. This is something we're happy to help with (arguably more at math.stackexchange than mathoverflow, although mathoverflow is in practice fairly forgiving for logic questions), but embedding philosophical concerns inside an unclear question, which keeps shifting its goalposts (often nonsensically) in the comments, doesn't yield a good question.

So that's why I downvoted.

My advice for you going forward is twofold. First, ask clear questions; in particular, if you're asking a question here you should have a solid understanding of the meanings of the basic terms in that question, so there shouldn't be any unclearness resulting from basic misuse. Second, make understanding the answer to your question your priority, rather than understanding how it fits in with your philosophical conceptions. These are two different tasks, and it's my experience that your eagerness to do the latter gets in the way of you doing the former.

• Noah, you may be right about there being philosophical agenda getting in the way here. But speaking as someone outside set theory, I think also there was just some simple honest confusion, and some of the tone I saw seemed a tad impatient about the confusion (I felt this personally when I tried to ask a question at the thread). I would like to quote Bill Thurston here, from his MO profile (continued next comment).
– Todd Trimble Mod
Aug 26, 2017 at 0:13
• "Mathematics is a process of staring hard enough with enough perseverence at at the fog of muddle and confusion to eventually break through to improved clarity. I'm happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication."
– Todd Trimble Mod
Aug 26, 2017 at 0:14
– Todd Trimble Mod
Aug 26, 2017 at 0:20
• @ToddTrimble "I think also there was just some simple honest confusion, and some of the tone I saw seemed a tad impatient about the confusion (I felt this personally when I tried to ask a question at the thread)." That is quite true and regrettable (especially the spillover to you). And I hope my own tone has been better. But at the same time, I stand by what I've said in my answer. In particular, I don't quite mean that there's a philosophical agenda here (although I do think there is); rather, the OP is interested more in doing philosophy than in understanding the mathematics here (cont'd) Aug 26, 2017 at 0:22
• Or at least, that's what comes through based on their engagement in this question (and in others). The confusions the OP is having are meaningful ones (incidentally in response to your comment let me just point out that I have in fact answered a large number of the OP's questions in the past - I don't think it's fair to paint me as intolerant of confusion in general or of the OP in particular); what I object to in this question, and in some others, is the way in which the OP goes about trying to resolve those confusions or responds to them being pointed out. (cont'd) Aug 26, 2017 at 0:33
• (I'd like to clarify, by the way, that here I am criticizing the OP's question - I am not trying to defend the tone of all participants involved. I can be frustrated by multiple things at once.) This is a difficult sort of thing to say, since it really can't help but risk having a chilling effect, but I think it's important for the OP to hear. This is not to say, by the way, that philosophy-heavy questions don't have their place on this site; as long as they have sufficient math content, I think they do. But I also think that to ask such questions well, one has to have very solid basics. Aug 26, 2017 at 0:40
• Fair enough. Actually, I didn't think of you as being intolerant here.
– Todd Trimble Mod
Aug 26, 2017 at 0:40
• (cont'd) Specifically, it's easy for logic questions to be somewhat inherently subjective (and I've asked plenty myself with large subjective components), but then the appropriateness of such a question on mathoverflow depends in part on the OP's ability to engage the issue of what they really mean. This is really my problem here: my impression from this question and elsewhere is that the kinds of question the OP really wants to ask are not ones they are currently able to ask in a productive way, and that trying to do so is not helpful. As a specific example from this question: (cont'd) Aug 26, 2017 at 0:43
• Subquestion (ii) is a good example of this for me. I don't understand what the hereditarily finite sets have to do with anything, or what notion of predicativity the OP is using, or (and this is the important one) what criteria they are using to judge what the "right" powerset notion of a theory should be. There's a mass of mathematical and philosophical content implicit here, and before we can even get to that we need to address the OP's question (i), which already I don't think is a question they are ready to ask at the moment. Now on some level I think I do know what's being asked here - Aug 26, 2017 at 0:47
• that in some sense $\mathcal{P}_{def}$, or rather the appropriate iteration of $\mathcal{P}_{def}$, gives us a notion of powerset sufficient for ZF - but I'm not confident that that's what is really being asked, since that is a trivial consequence of the consistency of ZFC+V=L, which the OP knows. Or rather, they know the statement of that fact, but I don't know if they really understand what it means and I have no better sense now than I did before of what the OP does and does not understand. I bring up philosophy, but leaving that aside: I don't think this question can have a happy ending. Aug 26, 2017 at 0:53
• @ToddTrimble Wrapping up, I think my tldr response to your comment is: (i) I think the question is a bad one both because it is fundamentally unclear and because I don't think that in the near future it can receive an answer the OP will find satisfying. (ii) I regret the tone of the discussion at the question, which as you quite correctly pointed out is at points unconstructive. (iii) I think that the advice the OP should take to heart is structural: to reframe the focus of their future questions, and work to master the basics so that they can ask clear versions of the questions they want. Aug 26, 2017 at 0:59
• I didn't get subquestion (ii) here either. But the question did have a happy ending in the small respect that I myself gained some understanding. :-) So extrapolating from that: it might be useful on occasion to think of addressing not just the OP directly but the community at large: what the issues seem to be, what the resolutions are. Speaking over the head of the OP to the crowd out there (to be polite to Thomas: not saying he's incapable of understanding, but rather that it's a community question). I get the sense this is what Joel David Hamkins is often doing: speaking to the community.
– Todd Trimble Mod
Aug 26, 2017 at 1:02
• It's also a good classroom technique: taking what is sometimes a suboptimal question and finding a useful lesson to extract for the world at large. All that preaching aside: you could well be right about the general trend of the OP's questions, and I hope he will reflect on everything you've said. I think this has been in many ways a useful discussion, and I thank you (again) for your honesty and forthrightness.
– Todd Trimble Mod
Aug 26, 2017 at 1:05