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The answer to this question would of course depend on what one means by "undergraduate real analysis". So my question, asked by a layman in logic myself, may be vague. You may consider it as the real analysis in the usual undergraduate curriculum. (Maybe it is more appropriate to replace "real analysis" with just "undergraduate math" or something else.)

I do "know" a bit of ZFC set theory and basic first-order logic. But I have not taken a course in the foundation of mathematics before, neither an advanced course in mathematical logic. So I apologize in advance if my question sounds silly to experts.

Do we have a simple example (axioms/definitions/theorems) in real analysis that can not be written in the language of first-order logic?

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    In what language? In the language of set theory, ALL undergraduate analysis can be written in first order logic (at least in theory). Indeed, we care about set theory (at least in part) because it lets you formalize the rest of mathematics. If you want to know what can be written in the language of, say, real closed fields, then completeness is already inexpressible. – HallaSurvivor Jan 13 '21 at 17:25
  • @HallaSurvivor: thank you for your comment. I'm not sure if I understand your question correctly, for "language", but I mean "language" in first-order logic, namely "formulas" in the link. –  Jan 13 '21 at 19:54
  • This answer says that "using ZFC we can interpret second-order logic" and this answer says "ZFC is a first-order theory".// So I naively expect that there should be examples that cannot be written in first-order logic formulas. –  Jan 13 '21 at 20:00
  • By "language", I mean the atomic symbols you're allowing. In "the language of set theory" the only atomic symbol you're allowed to use is $\in$. In the language of real closed fields, you allow only $0,1,+,\times,\leq$. The article you've linked doesn't allow function symbols, but that's a fairly minor quibble (you can add a relation $R_+(x,y,z) \iff x+y=z$). Again, in the langauge of set theory, you can do most things. In the language of real closed fields, there are many things you can't say. This is closely tied to the fact that set theory "interprets" SOL, as you've pointed out. – HallaSurvivor Jan 13 '21 at 20:01
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    Completeness Axiom of the linear order ≤ of the real numbers: $∀X([∃yX(y)∧∃z∀y(X(y)→y≤z)]→∃z∀y(∃u(X(u)∧y≤u)∨z≤y))$ – Mauro ALLEGRANZA Jan 15 '21 at 07:13
  • I am told by some purists that you cannot quantify over sets or functions in standard FOL. I do so anyway to no ill effect AFAICT. It is inevitable in real analysis. I haven't found it necessary to quantified over propositions and predicates. – Dan Christensen Jan 15 '21 at 15:38

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It is all depends on your context, where you are working in.

Let's say we are working in First order ZFC, our language includes all of the logical symbols, and in addition we have the one 2-ary relation symbol: ∈, and our axioms are, well the ZFC axioms.

In this settings, you can pretty much do everything there is in real analysis$^1$, both first and second order(and third an...), that is because ZFC is pretty strong, we can think about $ℝ$ as a special set we construct, then by looking at $\mathcal P(ℝ)$ and $\mathcal P\mathcal P(ℝ)$ we can talk about "high order real analysis".

Things starts to become more interesting when we are in the settings of real analysis, more specifically, let's look at the example of ordered fields:

Our language contains the logical symbols: $\cdot, +, <$(multiplication, addition, and order) and the axioms of ordered fields.

In this context we can find a lot of "natural" second order sentences, every sentence that quantify over subsets, over functions or over relations. For example: "for every subset of the field, the subset is either unbound(from above) or there exists a supremum to this subset", this is the "Least upper bound property", and it is a second order sentence.

It is a bit harder to show that this sentence cannot be written using first order sentence, but it is possible to prove that.

But wait, why in ZFC we are allowed to quantify over those subsets? Well, because we are not quantifying over subset of our universe, in the context of ordered fields, our universe is the field we are working in, but in ZFC our universe is some weird collection of a lot of sets, and just like in ordered fields, it is possible to find "natural" second order sentences that you cannot talk about without quantifying over arbitrary sub-collection of the universe, that is, you cannot talk about them in FOL in the context of ZFC. But even then, we can just take a stronger theory, such as NBG set theory, or TG set theory, which allow you to talk about second order ZFC using first order sentences.

$^1$by that I mean that we express most of the things we care about, it doesn't mean that there is nothing interesting to say about real analysis in a different context, other than ZFC.

ℋolo
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