@Weier The keyword is "computable analysis" and you will find good stuff.

@TimothyChow The pity is that so many people seem to think of these journals as "generalist" and also that they are a reliable judge of quality across mathematics. Since I am also predisposed against journal rankings generally, finding them harmful to the subject, I would prefer that we judge the papers themselves (journal impact factors, for instance, are now completely discredited as a reliable means of judging quality of individual articles). I don't feel compelled to help out the Annals. I also disagree with your suggestion below that logic has some kind of prestige problem to solve.

If we just start with a naked set, however, then truth would be trivial, and so we could do the whole construction in PA I think. Perhaps that is what you meant.

@Kimball I game some numbers already, but you can do the search query at MathSciNet yourself with: j:("Ann. of Math.") AND pc:(xx) y:[0 2024] and replace xx with your desired MSC code.

Yes, that in fact is the advice I give. We talk mainly about the logic journals and which ones might be suitable for their work; the other journals just don't matter that much. Personally, though I have over 100 published articles, I haven't ever submitted a paper to Annals or any of the other journals being mentioned here as top 5 or whatever. It just hasn't seemed important.

Are you seriously suggesting that the logic numbers in the Annals are accurately reflecting the contribution of logic to mathematics? I would find that absurd.

I'm not complaining about bias. Rather, my complaint is that the Annals just isn't important for all subjects. In logic, it is nearly irrelevant. (Perhaps it is because of historical lack of knowledge of logic by other mathematicians and journals that caused logic to develop its own journals and societies, and so there basically isn't a culture of publishing in the supposedly generalist journals.) There are over 80,000 articles on MathSciNet with subject code 03, but only 24 of them are in the Annals. To me, that failure reflects more on the Annals than it does on logic as a subject.

@pastebee I was imagining starting with a model of arithmetic, and then defining the $\in_n$ relations as an expansion of this. I don't quite see how to define the $\in_n$ relations in PA, since we'd need to decide which formulas are equivalent, which is why I think a truth predicate is involved. Can you explain your idea a bit more?

Meanwhile, 351 papers with pc:15 (algebraic geometry), and 311 papers with pc:11 (number theory).

For comparison, according to the search j:("Ann. of Math.") AND pc:(03) y:[0 2024] the Annals has published a grand total of 24 papers in logic, altogether since its inception (probably this misses papers that don't use the MSC). Logic is a huge field with more than a dozen of its own specialist journals, its own professional societies, and deep connections with essentially all parts of mathematics, and also deep connections with neighboring areas such as theoretical computer science.

I would write just $\phi$, but say it is part of the rank $n$ language if it has $\in_m$ only for $m<n$. That is, you are defining a hierarchy of languages, rather than putting labels on formulas.

What a pity that these supposedly top generalist journals are apparently so narrow. Perhaps I should start advising my students to switch to algebraic geometry or number theory, if they ever want to be real mathematicians.

@ZuhairAl-Johar I just meant that $n$ is not determined by $\phi$, since any larger $n$ also works, so the notation $\phi^n$ isn't very good.

No, it is not strong enough (but I don't know anything about TST). If you start with $X$ as the standard model of arithmetic and do my construction on top, then this is rather low in the hyperarithmetic hierarchy, since you are just iterating a truth predicate $\omega$ many times. There is a model computable from $0^{(\omega^2)}$.

I think that is the right way to do it, and that is how my answer takes it. This means that the superscript on $\phi^n$, however, could also be anything larger than that.