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In a prior posting here, the following is a quote from an answer:

$\mathsf{ZFC}$ is so ridiculously overpowered that its inconsistency wouldn't really spill over to the rest of mathematics too much. For the vast majority of mathematical practice, galactically weaker theories like $\mathsf{ZC}$ are enough.

, and it continues to say:

It turns out that a huge amount of mathematics can be developed in $\Pi^1_1 \mathsf{CA_0}$, which is a tiny fragment of $\mathsf{Z_2}$ which is itself a tiny fragment of $\mathsf{ZC}$

My question:

If $\mathsf{ZFC}$ is so ridiculously overpowered (over the goal of formulating mathematics), then why its largely held as the foundation of mathematics? Why should I build a skyscraper just for me to enjoy the shade at summer?

Why not $\mathsf{Z_2}$, or Pocket set theory, or even going high up to MacLane set theory? That seems to be more reasonable. Then we can consider further extensions of those only on demand i.e. when we need to formulate the very scantly amount of mathematics that requires strong foundations, and that would never end.

Asaf Karagila
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Zuhair
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    Personally, I'm fine with teaching everyone the most powerful and comprehensive language possible and then have some specialists reverse-engineer the results to see if they fit in some weaker theory, if the alternative is to teach everyone a language that makes topology and measure theory extremely hard to treat, only to give a custom-made extension for the specialists of those field. Hard open problems need everything to be at disposal, and then some. –  May 05 '20 at 10:08
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    A first guess would be : it's a historical artefact. A second guess would be that sometimes you use replacement when you actually only need comprehension, but you'd have to think about it, about how to turn what you did into something that's acceptable within the weaker system - which is most likely doable most of the time. So having all the strength from ZFC allows you not to have to think about it, and so to focus on the actual mathematics rather than the set theoretic issues (if you're not doing set theory - of course if you are then it's a different story) – Maxime Ramzi May 05 '20 at 10:09
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    While I don't know what those alternate theories are, my guess is that ZFC is simply the most practical theory to work in. – celtschk May 05 '20 at 10:11
  • The language of set theory is the "foundational" part. Plus some specific axioms needed in some branches of mathematics: e.g. AC for analysis. – Mauro ALLEGRANZA May 05 '20 at 10:35
  • https://math.stackexchange.com/questions/2598989/why-did-we-settle-for-zfc and https://math.stackexchange.com/questions/2130269/why-did-mathematicians-choose-zfc-set-theory-over-russells-type-theory and almost definitely a few other questions before these. – Asaf Karagila May 05 '20 at 11:18
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    How can you talk about arbitrary Banach spaces in Pocket set theory? You only have those that have size continuum. And even they don't exist as objects in your theory per se. How do you even talk about things like arbitrary rings or groups? – Asaf Karagila May 05 '20 at 11:21
  • @AsafKaragila what consistency level it takes to speak about those arbitrary Banach spaces? – Zuhair May 05 '20 at 15:31
  • @MauroALLEGRANZA, I think this is too harsh reduction, to say that foundation is just FOL with equality plus the primitive of membership, and perhaps plus the axiom of Extensionality I suppose. And then add axioms as required. – Zuhair May 05 '20 at 18:41

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I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories.

Perhaps one of the most significant advances in foundations is the identification of the consistency strength hierarchy. It allows us to calibrate mathematical statements with "canonical" extensions (guided by large cardinal axioms) or restrictions of ZFC. The former has been of greater interest to set theorists, so it is more visible, but the restrictions are just as important. They allow us to calibrate all sorts of things, from fragments of definable determinacy, to questions of analysis or bounded arithmetic.

In practice, many mathematicians (including many set theorists) only work in one of these restrictions, a small (or even very small) fragment of ZFC. If such a fragment suffices for your purposes, that's the one for you to use. In fact, I would argue that you should strive to work in the fragment that is appropriate for your mathematical needs, using only additional axioms beyond that fragment as your mathematics demands it.

On the other hand, some of us are interested in questions that demand significant strength beyond ZFC. For us, the appropriate theory to work on is an extension of ZFC (or NBG, or MK) with large cardinals, or a theory equiconsistent with such an extension, or a theory expected to be equiconsistent with such an extension. But also, sometimes the questions we want to look at are essentially combinatorial and $\mathsf Z_2$ (or an even weaker fragment) is the right framework in that case.

What matters is that we can go as high or as low within the ladder of the consistency strength hierarchy as the mathematical problems we encounter demand us without having to keep switching frames, so that the set-theoretic scaffolding is in many cases in the background, and only remarked upon when its presence is relevant. (For instance, questions about generic absoluteness demand constant attention to the set-theoretic framework. Questions about the partition calculus may only require a working knowledge of it.)

Andrés E. Caicedo
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  • if you look at what you wrote you'll see that ZFC occupies some special position in this ladder of consistency, for instances you are dividing theories into extensions of it and into fragments of it, so it has some key position! Why it is so? you could have taken some hierarchy theory whose stages are below $V_{\omega+\omega}$ and consider it as the key theory and divide matters into those extending it and those that are fragments of it. And actually if you are thinking of set theory as gauging consistency strengths, then a hierarchy theory rather than ZFC would suit more such intention. – Zuhair May 05 '20 at 17:02
  • @Zuhair "Why it is so?" Simply because it has a name, and historically it was significant and so it is easy to present and refer to it in texts, particularly texts where we do not care how much reflection we use or whether choice or power set or replacement is used in some specific instance. What is significant is the whole hierarchy. Think of it as a ladder. ZFC is one of its rungs. Any of its predecessors is a restriction, anything beyond is an extension. It is just convenient. (Now, not everybody agrees with this description, of course. Shelah, for example, regards ZFC as significant.) – Andrés E. Caicedo May 05 '20 at 17:54
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    I don't have any problem with this characterization at all, however one can capture this rationale more by directly coining hierarchy theories, like the one presented at: https://math.stackexchange.com/questions/3654927/is-zfc-a-seamless-embodiment-of-the-notion-of-a-hierarchy?noredirect=1#comment7512527_3654927, and then simply name each of those as $H_\alpha$ where $\alpha$ is the lowest ordinal it fails to capture, so the theory mentioned above can be called $H_\theta$ where $\theta$ is the first fixed point on the omega function. So the $H_\alpha$'s are the foundation of mathematics. – Zuhair May 05 '20 at 18:00
  • I think it is a matter of preference and presentation style, at this point. – Andrés E. Caicedo May 05 '20 at 18:08
  • What was Shelah's rationale? – Zuhair May 05 '20 at 18:31
  • I don't know that I quite understand it, honestly. He talks about it here. The paper appeared as MR1234276 (94i:03002). Shelah, Saharon. The future of set theory. In Set theory of the reals (Ramat Gan, 1991), H. Judah, ed., 1–12, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993. – Andrés E. Caicedo May 05 '20 at 18:40
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    Thanks a lot for the answer, comments and this reference! – Zuhair May 05 '20 at 18:45