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A conclusion is formally derivable from premises when a permitted formal proof exists with those premises as its undischarged assumptions. This is a syntactic relation, distinct from semantic entailment.

A Hilbert proof system specifies axiom schemes and inference rules, commonly including modus ponens. A formal proof is a finite sequence of axiom instances, permitted assumptions and conclusions obtained from preceding lines by those rules. Soundness and completeness compare this syntactic derivability with semantic validity or semantic entailment. These properties concern the particular chosen axioms and rules.

For an independent axiom set, semantic entailment of any one axiom from the remaining axioms fails. By the completeness theorem, this is equivalent to no member being formally derivable from the others in a complete calculus. To establish independence it suffices to give, for each axiom, a first-order structure satisfying all the other axioms and failing that one.

Entail is co-NP-complete under polynomial-time many-one reductions. For valid propositional formulas, semantic entailment holds exactly when the logical conjunction of the premise and the conclusion's logical negation is an unsatisfiable Boolean formula:A counterexample to semantic entailment is a Boolean valuation making $\phi$ true and $\psi$ false. Guessing one bit for each distinct Boolean variable and evaluating both propositional formulas is a polynomial-time algorithm. Thus the complement of Entail belongs to NP. Malformed descriptions, also in that complement, can be accepted after a deterministic syntax check. Hence Entail belongs to co-NP as a formal language over all strings.

For hardness, the unsatisfiable Boolean formula problem is co-NP-hard: if $A\in \mathrm{co}\text{-}\mathrm{NP}$, then $\overline{A}\in \mathrm{NP}$, and the permitted NP-hardness of SAT supplies a polynomial-time many-one reduction $r$ withWe may ensure $r(x)$ is always a valid propositional formula, replacing any malformed output by a fixed contradiction. It follows that $x\in A$ exactly when $r(x)$ is unsatisfiable. This justifies using the unsatisfiability problem as a hardness source without an extra assumption.

Map a valid propositional formula $\theta$ towhere $p$ is a fixed Boolean variable. The second propositional formula is false under every Boolean valuation. Consequently it is entailed by $\theta$ precisely when no Boolean valuation satisfies $\theta$. ThusThe output has only constant additional length and is computable in polynomial time. If the source unsatisfiability language rejects malformed strings, map them to the fixed no-instance $⟨p,p\wedge \neg p⟩$. This completes a total polynomial-time many-one reduction and proves the claimed co-NP-completeness.

Let $S=\{F_i:i\ge 1\}$. Every finite subset of $S$ is satisfiable: if $F_N$ has the largest index occurring, the successive semantic entailments imply $F_N\models F_i$ for every earlier index, so any Boolean valuation satisfying $F_N$ satisfies that finite subset. The propositional compactness theorem therefore also shows that $S$ itself is satisfiable.

Suppose that $G$ had exactly the common models of $S$. Then $S\models G$, and hence $S\cup \{\neg G\}$ is unsatisfiable. By the propositional compactness theorem, finitely many $F_i$ together with $\neg G$ are already unsatisfiable. Enlarge the finite list if necessary so it is nonempty, and let $N$ be its greatest index. We obtainOn the other hand, the asserted equality of model classes gives $G\models F_{N+1}$. Transitivity of semantic entailment would yieldcontradicting the strictness assumption at index $N$. ThusThe obstruction is the finite character of propositional formulas: a single consequence of an infinite set already follows from finitely many premises, whereas this chain continues to impose strictly stronger requirements forever.

A first-order theory in a fixed first-order signature is a set of first-order sentences. Under the closed-theory convention used in the question, it contains all its semantic entailments: $T\models \theta$ implies $\theta \in T$. One can equivalently begin with a set of axioms and take its closure under consequence.

A complete theory is a consistent first-order theory that decides every first-order sentence $\theta$ in its first-order signature:Consistency ensures that exactly one of these alternatives holds.

A decidable theory is one for which an algorithm always terminates and determines whether a given first-order sentence is a consequence:If the first-order theory is already closed under consequence, this is its membership problem. It is a condition on the consequences, not merely on whether the chosen set of axioms has a decidable membership test.

For independence, construct four first-order structures, each satisfying exactly the three required remaining axioms and failing the chosen one.

To fail associativity, use the carrier $\{0,1,2\}$, interpret $c$ as $0$, and define the commutative operation by its table:The first row and column give a two-sided identity, and every element is its own inverse because the diagonal entries are $0$. Symmetry gives commutativity. Butso associativity fails.

To fail the identity axiom, take $\{0,1\}$, $c=0$, and the constant operation $F(x,y)=0$. It is associative and commutative, and every element has a two-sided partner producing $c$; but $F(1,c)=0\ne 1$.

To fail the inverse axiom, use natural numbers under addition with $c=0$. Addition is associative and commutative and has the required identity, but $1$ has no additive inverse in this carrier.

To fail commutativity, use the symmetric group $S_3$ under permutation composition with $c$ its identity. Associativity, identity and inverses are group properties, while $(12)(23)\ne (23)(12)$. Each countermodel witnesses the failure of semantic entailment from the other three to the failed first-order sentence. Hence the four source axioms are independent.

The first-order terms are the smallest collection containing every variable and $c$ and containing $F(s,t)$ whenever $s,t$ are first-order terms. The atomic formulas are exactly $s\approx t$ for first-order terms $s,t$. Logical formulas are then obtained from atomic formulas by the logical connectives and quantifiers: if $\theta ,\psi$ are logical formulas and $x$ a variable, then $\neg \theta$, $(\theta \wedge \psi )$, $(\theta \vee \psi )$, $(\theta \to \psi )$, $\forall x\theta$ and $\exists x\theta$ are logical formulas. These closure rules, with no additional strings admitted, define the syntax; a presentation using fewer primitive connectives treats the others as abbreviations.

Semantic entailment is the conditionThe same first-order variable assignment is used for all premises and the conclusion. This matters when free variables occur; for first-order sentences the truth value is independent of the first-order variable assignment.

Soundness states that formal derivability implies semantic entailment:For first-order sentences this means that every first-order structure satisfying all assumptions also satisfies every first-order sentence formally derived from them. In the logical formula version one uses the usual restrictions on generalization from assumptions with free variables.

Strong completeness states the converse:Equivalently, every syntactically consistent set of first-order sentences has a first-order model. Equality is interpreted as actual identity, and its usual logical axioms and substitution rules are part of the standard calculus. Both assertions hold for arbitrary sets of assumptions, rather than only finite sets.

For a standard classical propositional calculus, strong completeness statesTogether with soundness, this identifies semantic consequence with formal derivability. One proves the contrapositive by the maximal consistent set in propositional logic construction. If $\Gamma ⊬\varphi$, the set $\Gamma \cup \{\neg \varphi \}$ is propositionally consistent, by the deduction theorem for propositional logic and classical propositional reasoning. Extend it to a maximal consistent set in propositional logic $M$, deciding each logical formula in turn. At each stage at least one selection of $\theta$ or $\neg \theta$ preserves propositional consistency; a contradiction in the union would use only finitely many assumptions and hence already occur at a stage.

Define a Boolean valuation by $v(p_i)=1$ exactly when $p_i\in M$. Maximal consistency in propositional logic gives $\neg \theta \in M$ exactly when $\theta \notin M$, and $(\theta \to \eta )\in M$ exactly when either $\theta \notin M$ or $\eta \in M$. Structural induction on logical formulas therefore proves the maximal consistent set truth lemma $v(\theta )=1⟺\theta \in M$. Thus $v$ satisfies $\Gamma$ but falsifies $\varphi$, contradicting semantic entailment. The countable language here permits an enumeration of logical formulas; the corresponding general-language argument uses a well-ordering or a principle of maximality in a partially ordered set.

The assertion $\Gamma ⊢\varphi$ means that there is a finite formal derivation ending in $\varphi$ with its undischarged assumptions contained in $\Gamma$. In a context-labelled presentation, an axiom instance is available with empty context, a formula $\alpha$ is available from the context $\{\alpha \}$, modus ponens combines derivations of $\alpha$ and $\alpha \to \beta$ to derive $\beta$, and generalization derives $\forall x\alpha$ provided $x$ is not free in its open assumptions. Thinning enlarges a context without changing its conclusion.

Each line or node must specify an axiom instance, an assumption, or an application of a permitted rule to earlier lines or subderivations. The side conditions on substitution, quantification and contexts are part of correctness. In particular a derivation is a finite syntactic object, even when $\Gamma$ is infinite; it uses only finitely many assumptions before a final thinning to $\Gamma$ if necessary. Derivability $⊢$ is thus distinct in definition from semantic entailment $\models$, which quantifies over structures and assignments.

Adequacy here means functional completeness: every Boolean truth function of finitely many positive-arity propositional inputs can be expressed by a formula using the available connectives. It is an expressive-power claim, distinct from the completeness theorem relating derivability and semantic entailment.

Negation and implication define conjunction and disjunction byGiven any truth table, for each input row where its output is true form the minterm containing each input positively or negatively according to that row. The disjunction of these minterms is true on exactly the required rows: a disjunctive normal form. Replacing its conjunctions and disjunctions by the displayed definitions gives a formula using only $\neg ,\to$.

The constantly true and constantly false functions are represented by $p\to p$ and $\neg (p\to p)$, using a dummy input if needed. Thus the empty or full truth table also causes no difficulty. If one insists on genuinely nullary formulas with no variables at all, a separate truth constant is needed; ordinary propositional adequacy allows the constant truth functions to be expressed with irrelevant input variables.

For each term $t$ define an $L_2$-formula $E_t(z)$ expressing its value. Use fresh bound variables at every step:Freshness includes avoiding all relevant free variables of the formula being translated. Define the atomic translation byand extend through the logical operations: $\sigma (\neg \theta )=\neg \sigma (\theta )$, $\sigma (\theta \to \eta )=\sigma (\theta )\to \sigma (\eta )$, and $\sigma (\forall x\theta )=\forall x\sigma (\theta )$. Fresh witnesses prevent variable capture and ensure the translation preserves the original free-variable set.

For an $L_1$-structure $A$, let $A^r$ have the graph relation of $f^A$ and singleton relation $\{c^A\}$. Induction on terms proves that $E_t(z)$ holds precisely when $z$ is the value of $t$. Induction on formulas then provesConversely every $L_2$-structure satisfying $\varphi ,\psi$ is uniquely the relational version of an $L_1$-structure on that same domain. Hence, for arbitrary possibly infinite $\Gamma$, semantic entailment on the left is equivalent to semantic entailment from $\sigma (\Gamma )\cup \{\varphi ,\psi \}$ on the right, with assignments retained in the free-formula case. Applying soundness theorem for first-order logic and Gödel's completeness theorem givesThe prohibition on completeness in the preceding direct equality question is local to that proof; here it supplies the clean semantic-to-syntactic transfer after the actual translation and model correspondence have been established.

Totality of the binary relation is expressed byWhen $R$ is already an order, this adds comparability and makes it a total order. In fact this first-order sentence alone entails $\rho =\forall xR(x,x)$.

By the definition of semantic entailment, $\tau \models \rho$ means that every first-order structure satisfying $\tau$ also satisfies $\rho$. Let $\mathcal{M}$ be any such first-order structure, and let $a$ be any element of its domain of a first-order structure. The instance of $\tau$ with $x=y=a$ gives $R(a,a)\vee R(a,a)$, which is true only if $R(a,a)$ is true. Since $a$ was arbitrary, $\mathcal{M}\models \forall xR(x,x)$. ThereforeThere is no exception for $x=y$ in the displayed totality axiom; that case is exactly what forces the reflexive relation property.

A propositional formula $\phi$ semantically entails $\psi$ when every Boolean valuation satisfying $\phi$ satisfies $\psi$. Equivalently, $\phi \wedge \neg \psi$ is an unsatisfiable Boolean formula. Deciding semantic entailment is co-NP-complete: a counterexample Boolean valuation certifies failure in polynomial time, while $\theta \mapsto ⟨\theta ,p\wedge \neg p⟩$ reduces unsatisfiability to semantic entailment.