I'm studying model theory (after completing Naive Set Theory of Halmos) and I would like to understand the relationship between ZFC theory and an arbitrary theory in order to understand more the foundational characteristic of ZFC. Here I focus on ZFC and theory of Real Closed Field (RCF).
I would like to explain what I think about it and I'm more than happy to receive your critisism/enlightments.
General understanding of basic notation of Model theory
A language is a set of different kind of symbols (relation symbols, function symbols, logical symbols, etc). A theory is a set of sentences written in a language. As for me, a theory doesn't have anything "so precise". By saying that I mean a theory expresses the general ideas about how things work while leaving us to interpret them in a precise structure.
For example, in the formal language of ZFC, the axiom of empty set is :
${\displaystyle \exists x\,\forall y\,\lnot (y\in x)}$
which for me means that: There exists "something" called $x$ that there is no $y$ that "has relation" with $x$.
Here I think I can't say that $\in$ is the usual "belong to" relation. It's just the symbol (placeholder) for a relation that we will assign a precise meaning in a specific model of ZFC. I can't either interpret $x$ or $y$ as a set because there isn't any notion of set in the language of ZFC. Therefore, I called them "something".
Please stop me if I'm wrong!
ZFC theory and Real closed field theory
From the fantastic and elegant answer of Alex Kruckman: How to work with many theories at the same time?, I understand that ZFC theory is usually chosen as the foundation of mathematics because it can be used to define structures that eventually also satisfies other theories.
In one of his comment, Alex said :
"For each axiom $𝜑$ of RCF, it is a theorem of ZFC that $𝜑$ is true
in the field $\mathbb{R}$."
In fact, I want to understand the relationship of ZFC and RCF without having in mind a specific model of ZFC and normally, to prove a theorem of ZFC, we don't use a specific model (we use directly the axioms expressed in its language), as stated in the answers of tomasz and Tanner Swett (To prove a theorem of a theory, is it necessary to prove it is true in every model of the theory?)
Alex claimed also that:
"Nothing I've said in any way relies on having in mind a specific
model of ZFC"
Here I don't understand very well because by claiming the theorem "$𝜑$ is true in the field $\mathbb{R}$", I see that we are provoking $\mathbb{R}$ which is defined in a specific model of ZFC consisting of (sets, $\in$) (say, I believe it is a model of ZFC).
$\mathbb{R}$ for me doesn't exist in the formal theory of ZFC (i.e. it exists as just "something" in the language of ZFC, waiting to be interpreted/defined in a precise model). So, I think we are indeed placing ourselves in a model of ZFC while claiming like the first statement of Alex. Moreover, a statement is true for a model isn't necessarily true for every model (hence the whole theory) of ZFC. Please correct me if I'm wrong!
Questions
I would like to have 3 small questions please (the 2 first ones are related):
- When looking at the first statement of Alex, in which "environment" are we working ? In a specific model of ZFC, or directly in the ZFC theory/language or something else?
- If it is directly in ZFC formal language, how should I think about $\mathbb{R}$ ? Does $\mathbb{R}$ actually have a precise meaning within the ZFC theory (i.e. formal language) or it is just a "notion" waiting to be interpreted in a model ?
- Can I understand that for every axiom of RCF (written in the language of RCF), there is an equivalent theorem of ZFC (written in the language of ZFC) ? Therefore, can I think that RCF theory a "subset" of ZFC ?
The motivation for the 3rd question is that $\mathbb{R}$ is "governed" by ZFC while at the same time being a Real Closed Field, therefore I think ZFC should agree with RCF theory perfectly. And by saying "subset", I mean that ZFC has the ability to say much more that RCF theory and it can actually say everything (with the version in its own language) that RCF theory can say.
Thank you very much for your help!
P/S: I know it is off-topic but I want to thank the community of MathExchange for your kindness and for the values you are bringing to me, especially as a working person loving math with limited access to academic support. And sorry for the lengthy question!
Best Answer
If you are using ZFC as a foundational theory, you typically assume that all of mathematics happens inside a fixed "standard" model $V$ of ZFC. Whether you consider that to be literally true or just a convenient metaphor or something else is really a question about philosophy more than mathematics.
Regarding your second question, in the context of ZFC as a foundation (that is, given that you take for granted some fixed set-theoretic universe $V$ in the background), $\mathbf R$ has a precise meaning (well, precise up to the choice of a specific construction you choose; however, the results of these constructions are all definably (in V) isomorphic).
However, the fact that ZFC proves that $\mathbf R$ satisfies the RCF axioms is not really dependent on that $V$. What happens is that the field of real numbers is definable in ZF. More precisely, there is a formula $\rho(x)$ in the language of set theory (given by the construction!) which in the "standard model" of set theory defines exactly the field of real numbers.
Then ZF proves that there is a unique set satisfying $\rho(x)$ and given an axiom $\varphi$ of RCF, we can write a formula $\varphi'(x)$ in the language of set theory such which expresses $x\models\varphi$, and for each such axiom, it is a theorem of ZF that $\rho(x)\rightarrow \varphi'(x)$ (in fact, ZF proves much more than the first-order properties: it also proves that $\rho(x)$ implies that $x$ is complete, which defines it up to isomorphism).
Thus, $\mathbf R$ is also a notion that can be interpreted in any model of ZF (via $\rho$), and it will have the same first-order properties in each case.