# Atomic and elementary diagram of an $\mathcal L$-structure

I have a question motivated by $$\textbf{Definition 2.3.2}$$ of Marker's Model Theory: An Introduction.

Definition. Suppose that $$\mathcal M$$ is an $$\mathcal L$$-structure. Let $$\mathcal L_M$$ be the language where we add to $$\mathcal L$$ constant symbols $$m$$ for each element of $$M$$.

The atomic diagram of $$\mathcal M$$ is $$\{\phi(m_1,\dots,m_n): \phi \text{ is either an atomic } \\ \text{ \mathcal L-formula or the negation of an atomic \mathcal L-formula and \mathcal M\models \phi(m_1,\dots,m_n)}\}.$$

The elementary diagram of $$\mathcal M$$ is $$\{\phi(m_1,\dots,m_n): M\models \phi(m_1,\dots,m_n),\phi \text{ is an \mathcal L-formula}\}$$.

We let $$\operatorname{Diag}(\mathcal M)$$ and $$\operatorname{Diag_{el}}(\mathcal M)$$ denote the atomic and elementary atomic diagrams of $$\mathcal M$$, respectively.

Is the language $$\mathcal L_M$$ used somehow in these definitions? If so, how exactly? I don't see it.

Also, all these definitions look completely unmotivated and incomprehensible. What does lie behind these definitions? Is there something one can say in order to make them more plausible? Otherwise for me they look like a set of words, which I cannot make sense of.

• Please make your questions self-contained rather than linking to images. Commented Oct 10, 2018 at 22:01
• I can understand why you might find the definition unmotivated, but incomprehensible? All the notation and terminology used here has been clearly defined earlier in the book. Can you be more precise about which parts you're unable to parse? Commented Oct 10, 2018 at 22:05
• @AlexKruckman I edited the question, sorry! From the formal point of view the definition is comprehensible, but when you don't know why the author gives such definition, it's hard to comprehend it. For example, the definition of a structure is motivated by giving examples, and once the examples have been given, it's clear that the definition of a structure is natural; there is no need to memorize it because that's the only way to define what we have in mind. But in the case of diagrams I have to memorize this definition because I have no idea what lies behind it. Commented Oct 10, 2018 at 22:14
• For the exact parts, as I said, I don't understand how $\mathcal L_M$ is used here. (And why would someone want to expand $\mathcal L$ by adding additional constants? It's also completely unmotivated.) Are the $m_j$ supposed to be the constants that have been added? Are there any very simple examples that can be given here? Commented Oct 10, 2018 at 22:19

Yes, the language $$\mathcal{L}_M$$ is used in writing down $$\phi(m_1,\dots,m_n)$$.

That is, if $$\phi(x_1,\dots,x_n)$$ is an $$\mathcal{L}$$-formula, it can't refer to elements of $$M$$ (except those named by constants). To express the fact that $$\varphi$$ is true of the tuple $$(m_1,\dots,m_n)$$ from $$M$$, let's add a constant symbol called $$m_i$$ for all $$1\leq i \leq n$$ and interpret $$m_i$$ in $$M$$ in the obvious way, namely as the element $$m_i$$. Now $$\phi(m_1,\dots,m_n)$$ is a sentence in the new, larger language. Formally, it's the sentence you get by substituting the constant symbol $$m_i$$ for the variable $$x_i$$ in $$\phi(x_1,\dots,x_n)$$.

If you add constant symbols in this way for every element of $$M$$, you can now write down all the first-order truths about tuples from $$M$$. This set of $$L_M$$-sentences is the elementary diagram of $$M$$. If you restrict your attention to atomic and negated atomic formulas, you get the atomic diagram of $$M$$.

For example, in the ring $$\mathbb{R}$$, $$\pi^2 \neq e$$ is a negated atomic $$\mathcal{L}_{\mathbb{R}}$$-sentence. There is no sentence in the language of rings which asserts that $$\pi$$ is not the square root of $$e$$. There's just no way to talk about these real numbers in the language of rings.

As for the motivation, the definition is preceded (this is from Marker's Model Theory: An Introduction, p. 44) by the sentence "Next we give a way to construct embeddings and elementary embeddings."

Ok, so the definition is supposed to be motivated by a desire to construct embeddings. After reading the definition, you might naturally wonder what it has to do with constructing embeddings.

Fortunately, this is answered by the very next lemma, which says that if $$N\models \text{Diag}(M)$$, then there is an $$\mathcal{L}$$-embedding $$M\to N$$, and if $$N\models \text{Diag}_{\text{el}}(M)$$, then there is an elementary embedding $$M\to N$$.

• Thanks! "To express the fact that $\varphi$ is true of the tuple $(m_1,\dots,m_n)$ from $M$, let's add a constant symbol $m_i\dots$" But given a structure $M$ and a formula $\varphi(x_1,\dots,x_n)$, one can interpret the tuple $(x_1,\dots,x_n)$ by assigning $(m_1,\dots,m_n)$ to it and then one can talk about $M\models \phi(m_1,\dots,m_n)$. That is, I don't understand why we need to add new constant symbols to express the fact that you refer to in the sentence I quoted (instead we can just use the definition of $M$ satisfying $\varphi(m_1,\dots,m_n)$ in $M$). Commented Oct 10, 2018 at 22:46
• Right, you can also express the fact that $M\models \phi(m_1,\dots,m_n)$ by interpreting the variables $x_1,\dots,x_n$ as $m_1,\dots,m_n$ in $M$. But now you have an $\mathcal{L}$-formula $\phi(x_1,\dots,x_n)$ together with an interpretation of the variables in $M$. The advantage of adding the constants is (1) you can just talk about $\mathcal{L}_M$-sentences, without keeping track of variable interpretations, (2) then you can all at once talk about all the first-order truths about all the tuples from $M$, while formulas only mention finitely many variables at at time, and ... Commented Oct 10, 2018 at 23:01
• (3) most importantly it's convenient to then talk about satisfaction of the (elementary) diagram in a totally different $\mathcal{L}_M$-structure $N$, as in the diagram lemma, Lemma 2.3.3. You could do all of this by introducing a coherent (infinite) collection of variables, one for every element of $M$, and talk about the interpretation of all those variables in $N$. But it's more convenient to introduce constant symbols. Commented Oct 10, 2018 at 23:02