Saturated Structures and the Completeness of Real Closed Fields

13 May 2025

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Let $\mathfrak{A}$ be an $\mathcal{L}$-structure. Let $A$ be the domain of $\mathfrak{A}$.

Definition. A subset $X \subseteq A$ is said to be definable if there exist some $\mathcal{L}$-formula $\phi$ and free variable assignment $g$ such that the following holds:

\[X = \{ x \in A : \mathfrak{A} \vDash \phi[g^0_x] \}\]

When $\phi$ has no free variables other than $v_0$, we say that $X$ is $\emptyset$-definable.

Remark. Definability has the same meaning as Gödel’s constructibility.

For example, in $(\mathbb{R}, <)$, the interval $(e, 2\pi)$ is definable by the following $\phi$ and $g$:

• $\phi \equiv (v_1 < v_0 \land v_0 < v_2)$
• $g: v_1 \mapsto e, v_2 \mapsto 2\pi$

However, $(e, 2\pi)$ is not $\emptyset$-definable, as there is no way to specify $e$ and $2\pi$ in $(\mathbb{R}, <)$.

On the other hand, in the standard arithmetic model $(\mathbb{N}, 0, +, S)$, the set of even numbers $E$ is $\emptyset$-definable by the following $\phi$:

• $\phi \equiv \exists y (x = y + y)$

All finite sets are definable. For instance, $A = \lbrace a_1, a_2, a_3 \rbrace $ is definable as follows:

• $\phi \equiv (v_0 = v_1) \lor (v_0 = v_2) \lor (v_0 = v_3)$
• $g: v_1 \mapsto a_1, v_2 \mapsto a_2, v_3 \mapsto a_3$

For the same reason, all cofinite sets are also definable.

In the previous post, we examined resilient families of sets. We now define the following:

Definition. Let $\kappa$ be an uncountable cardinal. We say that $\mathfrak{A}$ is $\kappa$-saturated if every collection of fewer than $\kappa$ definable subsets of $A$ is resilient. In particular, when $\mathfrak{A}$ is $|A|$-saturated, we say that $\mathfrak{A}$ is saturated.

Therefore, an $\aleph_1$-saturated structure $\mathfrak{A}$ is one in which, whenever a countable collection of definable subsets of $\mathfrak{A}$ satisfies the finite intersection property, their total intersection is also non-empty. Meanwhile, it is impossible for a structure $\mathfrak{A}$ to be $|A|^+$-saturated, since the following family of sets satisfies the finite intersection property but has empty intersection:

\[\Big\{ A - \{ a \} : a \in A \Big\}\]

The importance of saturated structures lies in the following theorem:

Theorem. Two elementarily equivalent saturated $\mathcal{L}$-structures of the same cardinality are isomorphic.

Proof. Omitted. The basic idea is a generalisation of Cantor’s back-and-forth argument seen in the previous post.

Unfortunately, saturated structures are difficult to construct. For instance, for an uncountable cardinal $\kappa$ and a consistent theory $T$ with $|T| \leq \kappa$, the theory $T$ has a $\kappa^+$-saturated model of cardinality $2^\kappa$. Therefore, under the generalised continuum hypothesis, such a model is saturated. However, it is known that ZFC alone cannot prove the existence of saturated structures.

We therefore introduce the following concept with a weaker condition:

Definition. We say that $\mathfrak{A}$ is special if $\mathfrak{A}$ is the direct limit of a directed system $\lbrace \mathfrak{A}_\kappa \rbrace _{\kappa < |A|}$, where $\kappa$ is an infinite cardinal and each $\mathfrak{A}_\kappa$ is $\kappa^+$-saturated.

Every saturated structure is special, as we can take each $\mathfrak{A}_\kappa$ to be itself. However, not every special structure is saturated. Therefore, being special is a strictly weaker condition than being saturated. Nevertheless, special structures satisfy the isomorphism property:

Theorem. Two elementarily equivalent special $\mathcal{L}$-structures of the same cardinality are isomorphic.

Moreover, special structures are easier to construct than saturated structures. In particular, the following specialisation of the Löwenheim-Skolem theorem is known:

Special Löwenheim-Skolem Theorem. Let $T$ be a theory in language $\mathcal{L}$.

1. If $T$ has an infinite model, then $T$ has a special model of cardinality greater than any given cardinal.
2. If $T$ has an infinite model and $\mathcal{L}$ is countable, then $T$ has a special model of cardinality $\beth_\omega$.

As an application of this theorem, we consider the following famous result:

Definition. The theory of real closed ordered fields or RCOF is a theory in the language $(0, 1, +, \cdot, <)$ consisting of the following axioms: ($x^n$ is an abbreviation for $x \underbrace{\cdot \;\cdots\; \cdot}_{n} x$)

1. Ordered field axioms
   • $\forall a, b, c : (a < b) \rightarrow (a + c < b + c)$
   • $\forall a, b : (a > 0 \land b > 0) \rightarrow ab > 0$
   • Field axioms
2. Square root axiom: $\forall a > 0 \; \exists x : x^2 = a$
3. Closure axiom schemas:
   • $\forall a_2, a_1, a_0 \; \exists x :x^3 + a_2x^2 + a_1x + a_0 = 0$
   • $\forall a_4, a_3, a_2, a_1, a_0\; \exists x : x^5 + a_4x^4 + \cdots + a_0 = 0$
   • …

The theory of real closed fields or RCF is a theory in the language $(0, 1, +, \cdot)$ consisting of the following axioms:

1. Field axioms
2. Formally real axiom: $\forall x : x^2 \neq -1$
3. Square root axiom: $\forall a \; \exists x : x^2 = a \lor x^2 = -a$
4. Closure axiom schemas

Tarski’s Theorem. RCOF and RCF are complete.

Proof. The key is the following lemma:

Erdős-Gillman-Henriksen Lemma. Two special real closed fields of the same cardinality are isomorphic.

Assuming the lemma, we prove Tarski’s theorem. If RCF were not complete, there would exist extensions $T_1$ and $T_2$ of RCF such that models of $T_1$ and models of $T_2$ are not elementarily equivalent. Since all models of RCF are infinite models (why?), by the special Löwenheim-Skolem theorem, $T_1$ and $T_2$ each have models $\mathfrak{A}_1, \mathfrak{A}_2$ of cardinality $\beth_\omega$. However, by the lemma, $\mathfrak{A}_1 \cong \mathfrak{A}_2$, which contradicts $\mathfrak{A}_1 \not\equiv \mathfrak{A}_2$. ■