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A canonical first-order structure is generated by closed-term names: each element is the interpretation of a closed term. In a first-order language without function symbols this says each element is named by a constant. The constant-name convention for canonical first-order structures agrees with this one in the function-free setting. Logical equality identifies different names for the same element.

A constant symbol is a zero-ary function symbol. Its interpretation in a first-order structure names one fixed domain element, and the symbol is a closed term. A language can contain no constant symbols unless a construction explicitly adds them.

An embedding of first-order structures is an injective function preserving function symbols and constants and both preserving and reflecting every relation symbol. Its image is a substructure. Satisfaction of an atomic diagram of a structure in an expansion gives such an embedding of first-order structures by sending each named element to its new interpretation; literal inclusion requires fixing the interpretations of those names.

No finite theory in the full language containing one function symbol for each scalar of an infinite field can axiomatize exactly its nonzero vector spaces. It mentions only finitely many symbols; changing an unused scalar function to violate preservation of zero leaves those finite sentences unchanged. Relative to the vector-space axiom schema, nonzeroness is instead the single axiom $\exists x(x\ne 0)$. Absolute finite axiomatizability and finite additional axiomatizability must be distinguished.

A function symbol is part of the syntax of a first-order language, with a specified finite arity. In a first-order structure it is interpreted as a function from the corresponding finite power of the domain into the domain. Applying it to the appropriate number of first-order terms forms another term. The symbol itself is distinguished from its interpretation in a particular structure.

The values of all closed terms in a first-order structure form a substructure when the first-order language has at least one constant. The constant ensures nonemptiness, and applying a function symbol to closed-term values again gives a closed-term value. Relations are restricted from the ambient first-order structure. Every universal sentence true in the ambient first-order structure remains true in this substructure.

An $L$-structure consists of a nonempty domain $A$, an element $c^{\mathcal{A}}\in A$ for each constant symbol, a function $f^{\mathcal{A}}:A^n\to A$ for each $n$-ary function symbol, and a relation $R^{\mathcal{A}}\subseteq A^n$ for each $n$-ary predicate symbol. When equality is part of the logic it is interpreted as actual identity on $A$.

An assignment in a first-order structure is a function $s$ from logical variables to $A$. It induces a term valuation of first-order terms recursively:For an atomic formula, substitute these first-order term values into the interpreted relation, or compare them for equality. Extend satisfaction for a set structure through the truth tables of the logical connectives, and through the quantifier clausesThus the first-order structure and assignment in a first-order structure together determine truth values for all logical formulas. A first-order sentence has no free variables, so its truth does not depend on the assignment in a first-order structure.

Use the following group axioms in the given language:Commutativity supplies the other identity and inverse equations. These first-order sentences therefore specify precisely an abelian group with the named identity $0$; an inverse function symbol is unnecessary because inverses are asserted to exist by a quantified formula.

For each positive integer $n$, let $t_n(x)$ be the first-order term recursively defined by $t_1(x)=x$ and $t_{n+1}(x)=t_n(x)+x$. Add the torsion-free abelian group axiom schemeEvery first-order model of these first-order sentences is an abelian group with no nonidentity element of finite order. Conversely every torsion-free abelian group satisfies each of the displayed first-order sentences. This is a countably infinite first-order axiomatization of exactly the required class.

Gödel's completeness theorem for first-order sentences says that, in a standard sound first-order Hilbert calculus,Equivalently every first-order consistent set of first-order sentences has a first-order model: apply this existence statement to $T\cup \{\neg \sigma \}$ for the consequence formulation.

For the first-order model existence formal proof use a Henkin construction. Enlarge the language by fresh constant symbols and adjoin Henkin witness axiomsFreshness of $c_{\theta }$ preserves first-order consistency. In detail, if a contradiction followed after adjoining one witness axiom, the deduction theorem for first-order logic would yield its negation, from which one obtains both $\exists x\theta (x)$ and $\neg \theta (c_{\theta })$ over the old theory. Since the new constant symbol does not occur in that theory or in $\theta$, replace it by a fresh logical variable and generalize to derive $\forall x\neg \theta (x)$, a contradiction. Finitely many witness additions preserve first-order consistency successively, and unions preserve first-order consistency because formal proofs are finite. Iterate the enlargement so that logical formulas involving newly introduced constant symbols also receive witnesses. Include at least one constant symbol to guarantee closed terms.

Extend the resulting first-order consistent witnessed first-order theory to a maximal consistent set of first-order sentences $H$. In the equality-free case make the domain the set of closed terms. Interpret constant symbols and function symbols by their formal first-order term constructions; interpret an atomic formula byThe truth lemma for a Henkin term model follows by induction on logical formulas with closed terms substituted for their free variables. The logical connective steps use maximal consistency of a first-order theory. In the existential step, membership of $\exists x\theta$ supplies a witness constant and hence a first-order term satisfying $\theta$; conversely a satisfying closed term gives the existential sentence by an inference of the calculus. The term model consequently satisfies $H$ and the original theory. Equality is absent here, so distinct closed terms may remain distinct domain elements without further identifications.

In the presence of equality, two syntactically different closed terms can be asserted equal by the Henkin theory, so the raw first-order term domain from part (b) must be replaced by a quotient set. DefineThe equality axioms give reflexivity, symmetry and transitivity, so this is an equivalence relation. Their substitution consequences make it a congruence relation on a first-order structure: replacing an input by an equivalent first-order term preserves equivalence of function values and preserves membership of atomic formulas in $H$.

Take domain $\{[t]:t\text{ is a closed term}\}$. Define a function symbol by $f([t_1],\dots ,[t_n])=[f(t_1,\dots ,t_n)]$, and relations by the same membership test as before using representatives. The congruence relation on a first-order structure makes these definitions independent of representatives. Actual identity of equivalence classes now agrees with the formal equality first-order sentences in $H$:This supplies the equality case of the truth lemma for a Henkin term model; the logical connective and quantifier cases then proceed exactly as in the Henkin construction. Thus the term model built from the quotient set is a genuine first-order model with equality interpreted as identity.

A first-order structure consists of a nonempty domain $A$, an interpretation $c^A\in A$ of each constant symbol, a total function $f^A:A^n\to A$ for each $n$-ary function symbol, and a relation $R^A\subseteq A^n$ for each $n$-ary predicate symbol. Logical equality is interpreted as actual identity.

An assignment in a first-order structure is a map $s$ from the variables into $A$. It extends recursively to terms: $x^A[s]=s(x)$, $c^A[s]=c^A$, and $f(t_1,\dots ,t_n{)}^A[s]=f^A(t_1^A[s],\dots ,t_n^A[s])$. Atomic truth is determined by membership of the resulting tuple in $R^A$, or equality of the two term values. Negation reverses the truth value, and an implication is false exactly when its antecedent is true and its consequent false.

For a universal quantifier,These clauses define a valuation on every formula by structural recursion. Only free-variable values affect a formula's truth; consequently a sentence has the same truth value under every assignment. This is a semantic assignment of domain elements, not a programming assignment operation.

Neither constants alone nor functions alone gives the same uniform adequacy on all nonempty carriers. A one-element set has exactly one interpretation of each function symbol of any finite arity and of every constant symbol. In contrast it has two interpretations of a unary predicate: empty or full. Thus a fixed purely functional or constant-only language has only one structure on that singleton, and cannot encode both truth values of $\exists xP(x)$ by a same-carrier truth-preserving translation with reconstruction. This is the singleton truth-preservation obstruction to expressive completeness, even when logical equality is available.

Constants alone also cannot represent general variable-dependent relations on larger infinite carriers: a fixed formula mentions only finitely many named elements, and its remaining equality tests do not encode an arbitrary binary relation. Functions offer more flexibility. On carriers with at least two elements, predicates can be encoded by characteristic functions taking two distinguished values, and constants can be represented by constant-valued unary functions if explicit constant symbols are forbidden. Such a construction can work after adding axioms for distinct truth markers, but those axioms exclude the singleton case. Enlarging the carrier or adding a Boolean sort is another possible encoding; it changes the same-carrier sense established in the preceding part and must not be silently identified with it.

Replace each $n$-ary function symbol by an $(n+1)$-ary graph predicate, with axioms asserting exactly one output for each input tuple. Replace each constant by a unary predicate asserted to hold of exactly one element. Recursively introduce witnesses for term values, then translate formulas through their logical connectives and quantifiers as above.

Every original structure has a unique corresponding relational structure satisfying these axioms, on exactly the same carrier, and every such relational structure reconstructs the original functions and constants. Truth under assignments and logical consequences are preserved in both directions. In this precise relative sense, purely relational languages are adequate to represent first-order languages with functions and constants: expressive power is retained after adjoining the graph and uniqueness axioms. Arbitrary relational interpretations that fail those axioms are not claimed to represent functions.

The alphabet of a first-order language separates logical symbols from nonlogical symbols. In the standard presentation here the logical symbols are the variables $x_0,x_1,\dots$, equality $\dot{=}$, negation $\neg$, logical implication $\to$, the universal quantification $\forall$, and the punctuation symbols parentheses and comma. The nonlogical symbols are constant symbols, function symbols of each specified finite arity, and predicate symbols of each specified finite arity, such as $f_n^{(k)}$ and $P_n^{(k)}$.

Constants can equivalently be treated as zero-ary function symbols. Their arities belong to the language specification. Other logical connectives, and $\exists x\varphi$ as an abbreviation for $\neg \forall x\neg \varphi$, need not be primitive alphabet symbols. This distinction matters in counting literal symbol occurrences.

The downward Löwenheim-Skolem theorem says that a structure in a countable language has a countable elementary substructure containing any specified finite or countable set of its elements. More generally the size bound is $\max (|\mathcal{L}|,|C|,{\aleph }_0)$ for a required parameter set $C$. In particular any satisfiable theory in a countable language has a model of size at most countable.

To see the last assertion directly from the completeness construction, add only countably many witness constants in the Henkin construction. There are then countably many finite closed terms, and taking their quotient cannot increase cardinality. The resulting term model is therefore countable. For the elementary-substructure version, perform the witness construction inside the given structure, choosing actual witnesses and closing under the function symbols. The induced substructure is elementary by the Tarski-Vaught test. This is the same witness mechanism as the completeness construction, with the ambient structure supplying the chosen witnesses.

The upward companion states that a theory with an infinite model has models in every sufficiently large infinite cardinal, at least $|\mathcal{L}|+{\aleph }_0$. Add that many pairwise unequal constants, apply compactness, and then the size-bounded construction. The countable consequence is the part needed below.

A prime model of a complete theory $T$ is a first-order model admitting an elementary embedding of first-order structures into every first-order model of $T$. Let $\mathcal{A}\models T$ be canonical and $\mathcal{B}\models T$. For each closed term $t$, defineEvery element of $A$ has a name, so this defines the desired map on all of $A$. If two first-order terms agree in $\mathcal{A}$, their logical equality first-order sentence holds in one first-order model of the complete theory and therefore in every first-order model of it; they agree in $\mathcal{B}$. This proves well-definedness. If they are distinct in $\mathcal{A}$, their inequality holds throughout the complete theory as well, proving injectivity.

For every logical formula $\varphi (\vec{x})$ and named tuple ${\vec{t}}^{\mathcal{A}}$, the substituted first-order sentence $\varphi (\vec{t})$ has the same truth value in the two first-order models by completeness of $T$. ConsequentlyThus $e$ is elementary. Since $\mathcal{B}$ was arbitrary, every such canonical first-order structure is prime. This formal proof also handles closed-term generation when function symbols are present; under the constant-name convention one simply uses constants as the first-order terms.

The set of closed-term values is nonempty because the first-order language has a constant. It contains the value of every constant. If $f$ is an $r$-ary function symbol and $t_1,\dots ,t_r$ are closed terms, thenwhich is another closed-term value. Hence the set is closed under every operation. Restrict the operations and relations of $\mathcal{A}$ to it; this defines a substructure $\mathcal{B}$ whose carrier is exactly the specified set.

Part (i) now applies to the universal first-order theory, givingIt is the substructure generated by the first-order language's constants, often called the ground-term substructure. The source's constant-symbol assumption ensures its nonemptiness, even when no other elements can be generated.

Here $\mathcal{B}\subseteq \mathcal{A}$ means a substructure, so the smaller carrier contains all named constants, is closed under every function symbol, and has each relation restricted from the larger first-order structure. First-order term evaluation on tuples from $B$ agrees in the two first-order structures. Consequently atomic formulas agree, and induction on Boolean connectives makes every quantifier-free formula agree as well.

Write an arbitrary universal sentence of $T$ as $\forall \vec{x}\theta (\vec{x})$ with quantifier-free matrix. Every tuple from $B$ is also a tuple from $A$, so $\theta$ holds at that tuple in $\mathcal{A}$. Quantifier-free agreement transfers it to $\mathcal{B}$. Since the tuple was arbitrary, $\mathcal{B}$ satisfies the universal sentence. ThusAn arbitrary subset without the substructure operations and relation restrictions would not be the hypothesis used by this formal proof.

An $L$-structure has a nonempty underlying set $A$. It assigns to each constant symbol $c_k$ an element $c_k^{\mathcal{A}}\in A$, to each $n$-ary function symbol $f_j$ a function $A^n\to A$, and to each $n$-ary predicate symbol $P_i$ a relation contained in $A^n$. Logical equality, when included as logical equality, is interpreted as actual identity.

The arities belong to the specification of the first-order language. An assignment of logical variables to elements of $A$ then evaluates first-order terms recursively and determines satisfaction of logical formulas through the logical connectives and quantifiers. A first-order sentence has no free variables, so its truth in the first-order structure is independent of the assignment.

For each fixed finite tuple length $n$, use the finite conjunctionIt asserts that no nontrivial scalar combination vanishes, exactly linear independence. It is an $L$-formula because the field is finite and every scalar multiplication is named. Scalar coefficients are indices of function symbols, not variables ranging over a separate field sort. There is no single finite formula with arbitrarily many free variables; the assertion is a formula for each prescribed finite $n$.