# Formally proving the consistency of a formal theory

This is based on a related question. I did some further reading to understand the problem. I'm just posting this question to check whether I have correctly understood the solution.

Let's say we are using ZFC as our meta-language (in standard first order logic). We have proven that a theory is consistent iff it has at least one model, and we have defined the satisfaction relation $$\models$$.

Now we wish to show that some formal theory $$\Gamma$$ is consistent. To do this, formally speaking, we would need to:

1. Define the constant, relation and function symbols as sets. Prove each definition describes a unique and existing object (using axioms of ZFC). Define another set $$\Gamma$$ containing the required axioms of our formal theory.

Also, by doing this, we have extended ZFC with definitions that makes $$\mathcal{L}(\Gamma)$$ a sub-language of extended-ZFC. This is required for the following.

2. Derive the axioms of $$\Gamma$$ as properties of the defined sets. The proof would be in the language of FOL and would be limited to the axioms of extended-ZFC.

3. Construct a tuple $$\mathfrak{M} = \langle \ldots \rangle$$ listing the constructed sets. This is a $$\Gamma$$-structure.

4. Using the definition of the satisfaction relation, prove that $$\mathfrak{M} \models \Gamma$$. The proof would again be in the language of FOL and would be limited to the axioms of extended-ZFC.

5. Thus, there exists a model of $$\Gamma$$. Therefore, it is consistent.

Am I right in thinking this is all we need to do, formally speaking? I know in practice we only need to do steps 1 and 2.

One important thing I learned is that, in this situation, we must pick a meta-language that allows us to talk about both $$\Gamma$$ and its model $$\mathfrak{M}$$.

Note: I am only looking to check my intuition. There may be better ways of proving consistency, but I am wondering whether my naive understanding could also work, and if not, what the problem is.

Update: I realized that one could regard ZFC as our theory instead, and simply define $$\Gamma$$-symbols in it. Then steps 1 and 2 are sufficient. But in doing so, ZFC is no longer a meta-theory, and so this solution addresses a different problem.

• It's definitely not necessary to change your meta-language or make any sort of extension of the language of ZFC to carry out this sort of consistency proof. Rather, if our meta-language is ZFC, then we must have some encoding of everything we want to talk about as sets. In particular, the symbols and formulas in the language of $\Gamma$ are encoded as sets (and this "Gödel coding" is much easier to do in set theory than in arithmetic, by the way). – Alex Kruckman Feb 7 '17 at 14:38
• @AlexKruckman Maybe I'm using "extension" incorrectly. What I meant is that you need to encode the symbols of $\Gamma$ as sets (that would amount to giving definitions right?) I was using "extension" in this sense, to mean we've defined some new symbols. I didn't know anything about Godel coding, so I read about it. This seems to be a different way proving consistency from what I wrote (no definitions, but numbering), right? Is this the easier way? Also, is what I wrote correct at least in principle? Thanks. – stranger Feb 7 '17 at 15:00

As Alex Kruckman writes in a comment, it is not necessary to change the language of ZFC. We do not need to prove $\Gamma$ in ZFC, and we do not need to specifically define the model $M$, we only need to prove that it exists.

So, working in ZFC, what we need to do is:

1. Write a formula $\phi(N,T)$ of ZFC which expresses "the set $N$ is a model of the theory $T$."

2. Prove that there is a set $M$ which is a model of our particular theory $\Gamma$. In other words, prove $(\exists M)\phi(M,\Gamma)$ for the $\phi$ in step 1. Note that we do not need to "define" $M$ or show that it is unique, we only need to prove that it exists. We do not need to extend the language of ZFC.

3. Prove that the model $M$ from step 2 cannot satisfy any contradictory sentence.

4. Show that if $\Gamma$ was inconsistent then the model $M$ from step 2 would need to satisfy some contradictory sentence.

Parts (1), (3), and (4) are standard results, which do not rely on $\Gamma$ in any special way. These are covered in any reasonable logic textbook, and I would just call them "formalizing first-order logic in ZFC". Only step (2) has to be adjusted to the particular theory $\Gamma$.

• Thanks. This is much neater, and I now see why you don't need to extend ZFC, since you're existentially "quantifying away" any particular model. – stranger Feb 11 '17 at 12:44