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

Question

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.

Answer 1

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?"

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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.

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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.

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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.

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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.