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An arithmetical proof relation represents the effective relation that a natural number encodes a formal proof of the logical formula with a specified Gödel number. With the usual coding apparatus its proof-checking relation is bounded. Existential quantification over the formal proof code gives the ordinary arithmetical provability predicate.

For an effectively axiomatized arithmetic theory, a arithmetical provability predicate expresses that a formal proof of the sentence with a specified code exists. If axioms are only enumerated, proof certificates also record finite stages at which their axioms appear. The resulting predicate is arithmetically expressible.

Map propositional letters to arithmetic sentences, map falsity to a refutable sentence, commute with implication and interpret $\square$ by an arithmetical provability predicate. Propositional axioms remain theorems. The K axiom uses the second of the derivability conditions, and the Löb axiom uses the Internal form of Löb's theorem. Modus ponens rule is preserved directly, and the necessitation rule uses the first derivability condition. Induction on a GL proof therefore proves the displayed soundness implication for Peano arithmetic with the stipulated provability predicate.

For adequate arithmetic provability predicates, the arithmetized deduction theorem identifies inconsistency of $S+\neg {\mathrm{Con}}_S$ with $S$-provability of ${\mathrm{Con}}_S$. Formalized second incompleteness gives ${\square }_S{\mathrm{Con}}_S\to {\square }_S\perp$. Contraposition yields the displayed conditional. This is an internal theorem and does not license removal of its consistency premise.

For a computably enumerable first-order theory $T$ extending Robinson arithmetic, choose the standard arithmetic arithmetical provability predicate ${\mathrm{Prov}}_T$. The diagonal lemma, carried out in the weak base theory, gives a sentence $G$ withIf $T$ is consistent, $T$ cannot prove $G$: an actual finite proof would be verifiable in the weak base, so $T$ would prove both its provability and its unprovability. There is therefore no actual proof of $G$, and the displayed equivalence, true in the standard natural numbers, shows that this canonical $G$ is true there.

If $T$ is omega-consistent, it cannot prove $\neg G$ either. Such a proof would imply the existence of a proof code for $G$, while each standard numeral can be individually verified not to be such a code. Ordinary consistency suffices for unprovability of $G$; the original argument for unprovability of its negation uses the stronger hypothesis. Requiring the fixed-point equivalence in a sound weak base is important: an arbitrary equivalence provable only in an unsound extension need not yield a true sentence.

For an arithmetical provability predicate write $\square A={\mathrm{Pr}}_T(┌A┐)$. The conditions are: if $T⊢A$ then $T⊢\square A$; $T⊢\square (A\to B)\to (\square A\to \square B)$; and $T⊢\square A\to \square \square A$. The first is an external rule, while the latter two are internal theorems. They formalize proof verification, closure under modus ponens rule and positive introspection. None licenses reflection from a proved sentence $\square A$ to $A$.

Starting from a recursively axiomatized arithmetic first-order theory $T_0$ with the usual arithmetical provability predicates, defineIf every stage is first-order consistent, the Gödel second incompleteness theorem makes every step strict in the consistency-strength preorder: the next stage proves the preceding stage's formal consistency statement, which that preceding stage cannot prove. Every formal proof from the effective union $T_{\omega }={\bigcup }_{n<\omega }{T}_n$ uses only finitely many axioms and lies in one stage. Thus the union is first-order consistent under this hypothesis and lies strictly above every finite stage. Arithmetic soundness of $T_0$ is a sufficient starting assumption, since each added formal consistency statement is then true in the standard natural numbers.

Consistency of $T_0$ alone does not ensure consistency of the next stage. Assuming Peano arithmetic is first-order consistent, $T_0=\mathrm{PA}+\neg \mathrm{Con}(\mathrm{PA})$ is first-order consistent by the Gödel second incompleteness theorem. Proof inclusion makes it prove $\neg \mathrm{Con}(T_0)$, so adding $\mathrm{Con}(T_0)$ produces an inconsistent next stage. The hypotheses for a progression must therefore distinguish a consistent starting theory from consistency of all its iterates.

Write $BA=P(┌A┐)$, with the corner notation understood as the numeral for the Gödel number. An arithmetical provability predicate numerates the theorems of $S$ in the standard natural numbers and satisfies the Hilbert-Bernays-Löb derivability conditions:The last two conditions are internal theorems, not rules authorizing reflection from $BA$ to $A$.

Here $P(x)=\exists y\mathrm{Prov}(x,y)$ numerates the theorem set by the given assumption. If $S⊢A$, its proof supplies a genuine natural-number witness, making $BA$ a true Sigma-1 formula. Sigma-1 completeness gives $S⊢BA$, proving the first condition. The second condition is supplied explicitly. Since $BA$ is itself a Sigma-1 formula, the given formalized Sigma-1 completeness yields $S⊢BA\to B(BA)$, the third condition. Thus the specified existential proof predicate is an arithmetical provability predicate.

Induct on a derivation in Gödel-Löb logic. Propositional axiom instances translate to theorems of Peano arithmetic. The image of $\perp$ is a refutable sentence, so it behaves as falsity in those translated propositional arguments.

The translation of axiom $\mathrm{K}$ is the distribution clause of the Hilbert-Bernays-Löb derivability conditions. The translation of axiom $\mathrm{L}$ isthe Internal form of Löb's theorem, whose derivation was given above and applies to the stipulated arithmetical provability predicate for Peano arithmetic. Thus every axiom has a provable translation.

The modus ponens rule is preserved because the interpretation commutes with implication. The necessitation rule is preserved by the first derivability condition: if $\mathrm{PA}⊢{\mathcal{I}}_i(A)$, thenHere the equality denotes the definition of the translated formula, not an arithmetic equality of truth values. A propositional substitution merely changes the arithmetic sentences assigned to the variables. The induction proves the Arithmetical soundness of GL:

A theory is recursively axiomatizable when it has an effective enumeration of axioms, equivalently when its deductive closure is computably enumerable. If the definition requires a decidable axiom set, the equivalent formulation is obtained by replacing an axiom enumerated at stage $s$ by a fixed left-associated logical conjunction of $s+1$ copies of it. To recognize a proposed padded axiom, inspect its finitely many possible decompositions into repeated conjuncts and check the corresponding finite enumeration stages. The padding preserves logical equivalence, so the resulting decidable axiom set generates the same theory. In either convention, enumerate formal proofs using the finite sets of axioms seen so far. This effectively enumerates every theorem.

Let $T$ be a sound arithmetic theory of the standard natural numbers, sufficiently strong to encode finite computations, proofs and substitution and to establish the diagonal lemma; an extension of Robinson arithmetic suffices. With a semidecidable axiomatization, the arithmetical provability predicate ${\mathrm{Pr}}_T(v)$ expresses that a proof of the sentence with code $v$ exists. A proof certificate can include enumeration stages for its axioms, so this predicate is arithmetically expressible even when the given axiom set is only computably enumerable.

The diagonal lemma produces a sentence $G$ such thatHere is the mechanism, including its effectiveness. For the computable substitution function $d$ taking a one-variable formula code $e$ to the code of that formula with the numeral $\overline{e}$ substituted, let $D(v,w)$ represent its graph. Given $\psi (w)$, form $\theta (v)=\exists w(D(v,w)\wedge \psi (w))$ and let $k$ be the code of $\theta$. The sentence $\theta (\overline{k})$ has code $d(k)$. Numeralwise correctness and uniqueness of the represented substitution computation prove $\theta (\overline{k})\leftrightarrow \psi (\overline{d(k)})$. This constructs a fixed point effectively from the code of $\psi$. Applying it to $\psi =\neg {\mathrm{Pr}}_T$ gives $G$.

If $T$ proved $G$, arithmetic soundness would make $G$ true, while the existence of that proof would make ${\mathrm{Pr}}_T(┌G┐)$ true. The displayed equivalence would then make $G$ false, a contradiction. Thus $T$ does not prove $G$. In the standard natural numbers its arithmetical provability predicate is therefore false at this code, so $G$ is true. Arithmetic soundness now rules out a proof of $\neg G$ as well. Hence $T$ is incomplete: neither $G$ nor $\neg G$ is a theorem, and $G$ is true.

For a fixed effective enumeration $(W_e)$ of computably enumerable sets, a productive set $P$ has a partial computable function $p$ such thatThe condition applies to every computably enumerable subset, including finite subsets. No condition is imposed on $p(e)$ when the inclusion fails.

Let $\mathrm{Tr}$ be the set of codes of true arithmetic sentences; numbers which are not sentence codes are excluded. Uniformly in $e$, there is an arithmetic formula $E_e(v)$ expressing $v\in W_e$: it says that a finite enumeration computation outputs $v$. Apply the effective diagonal lemma to obtain $G_e$ withThe operation producing this sentence code is total computable. Suppose $W_e\subseteq \mathrm{Tr}$. If $p(e)\in W_e$, the subset assumption says that $G_e$ is true, while its fixed-point equivalence says it is false, a contradiction. Thus $p(e)\notin W_e$. The same equivalence now says $G_e$ is true, so $p(e)\in \mathrm{Tr}$. We have provedTherefore arithmetic truth is a productive set. This proves productivity of arithmetic truth. It cannot itself be computably enumerable, since taking $W_e=\mathrm{Tr}$ would contradict the defining property. Applying the productive procedure to the enumerable theorem set of a sound arithmetic theory also gives a true sentence outside that theory. The productive construction does not require the enumerable subset of truths to be a theory or to be deductively closed.

Write $\square A$ for $P(┌A┐)$. A standard arithmetical provability predicate obeys the Hilbert-Bernays-Löb derivability conditions:The first condition is a rule about theorems; the latter two are statements proved within the first-order theory. When the terminology also includes external adequacy, require that $P(\bar{n})$ be true in the standard natural numbers exactly when the logical formula with code $n$ has an $S$-proof.

For Peano arithmetic, code finite strings, logical formulas and formal proof sequences by natural numbers. Formal proof checking is effective and has an arithmetized bounded checking relation with its computation data. Existentially assert the existence of a correct formal proof code and any required checking data. The resulting predicate symbol is Sigma-1, and its standard truth means exactly that an actual finite PA formal proof exists. Effective operations on formal proof codes combine an implication formal proof with an antecedent formal proof, and turn a formal proof into a formal proof that its code is a formal proof; PA verifies those constructions. They give distribution and positive introspection, while an actual theorem's finite formal proof gives its internal provability assertion. This is the requested rough construction, with the three internal formal proof properties distinguished from external agreement.

Use the arithmetical proof relation and coding just constructed. A formal proof of $C$ cannot have code $0$, since that code was reserved for $\neg C$. Every candidate code $v$ for a formal proof of $C$ is therefore positive. But code $0\le v$ is already a formal proof of $\neg C$, so the no-earlier-negation condition fails for every candidate. HencePA proves $C$ by induction, while $S$ has $\neg C$ as an added axiom. Thus $S$ is inconsistent and its ordinary arithmetical provability predicate is true of every first-order sentence, including $C$. The displayed negative Rosser assertion can nevertheless be true, because the Rosser condition compares the order of opposite formal proofs rather than asking for their mere existence.

There is an important coding qualification. The PDF refers to a course formal proof coding but does not specify its numerical ordering. The truth of this Rosser assertion for an inconsistent first-order theory is not coding-invariant, so formal proof length alone cannot establish it for an unspecified fixed coding. For example, a different legitimate numbering could reserve code $0$ for a known PA formal proof of $C$, code $1$ for the new axiom $\neg C$, and higher codes for all other derivations. Then code $0$ would witness ${\mathrm{Pr}}_S^{\ast }(┌C┐)$ instead. The explicit allowed construction in part (i) supplies a convention under which the requested assertion holds; interpreting it as a claim about some externally fixed course numbering requires that numbering's corresponding order property.

Use the Hilbert-Bernays-Löb derivability conditions for an arithmetical provability predicate. Abbreviate $P_S(┌A┐)$ by $PA$. The conditions needed below areThese are the usual theoremhood, distribution and positive-introspection conditions intended by a standard arithmetical provability predicate; arbitrary predicate symbols merely agreeing with provability externally need not satisfy them.

Since $S⊢\neg Y$, it proves $Y\to G$. The first two conditions give $S⊢PY\to PG$. For the converse, the fixed-point equivalence gives $S⊢G\to (PG\to Y)$. Applying theoremhood and distribution givesPositive introspection also gives $PG\to PPG$, and distribution combines these to yield $S⊢PG\to PY$. Thus $S⊢PG\leftrightarrow PY$. Combining with $G\leftrightarrow \neg PG$ provesIf $S$ proved $\neg PY$, it would prove $G$. The theoremhood condition would then give $PG$, while the fixed-point equivalence gives $\neg PG$. Therefore first-order consistency implies

Use the two formal consistency equivalences expressly supplied for this part. If $S⊢G\leftrightarrow R$, the derivability conditions give $S⊢\square \neg G\leftrightarrow \square \neg R$. Taking negations and applying those supplied equivalences yieldsOne-consistency implies ${\Sigma }_1$ soundness under bounded completeness. Hence $S+{\mathrm{Con}}_S$ is consistent: if not, $S$ would prove $\neg {\mathrm{Con}}_S$, a false ${\Sigma }_1$ sentence. But in that consistent extension the last equivalence and its consistency axiom prove its own consistency ${\mathrm{Con}}_{S+{\mathrm{Con}}_S}$. Gödel's second incompleteness theorem rules this out. ThereforeThis establishes the requested unprovability using the explicit formal hypotheses, without inferring internal provability from the semantic equivalence discussed in the preceding part. The hypotheses for the second theorem are the standard effective arithmetic presentation and adequate provability predicates stipulated here.

The fixed point of boxed negated provability gives, in fact, that GL proves $F\leftrightarrow \square \perp$. Here is a direct derivation. Since $F$ is provably equivalent to a boxed formula, positive introspection and necessitation of that equivalence give $F\to \square F$. Another introspection step gives $F\to \square \square F$. The stipulated fixed point also gives $F\to \square \neg \square F$. Combining these two boxed contradictories yields $F\to \square \perp$. Conversely $\square \perp \to \square \neg \square F$ follows by normal boxed explosion, and the fixed point then yields $\square \perp \to F$.

Arithmetical soundness of GL, proved by translating distribution into the second derivability condition, necessitation into the first, and Löb's axiom into its internal arithmetic theorem, now givesAny remaining propositional parameters may be assigned arbitrary arithmetic sentences; this equivalence is uniform in those assignments. If PA is one-consistent, it is consistent and ${\Sigma }_1$ sound. Its standard provability predicate for a contradiction is then a false ${\Sigma }_1$ sentence and cannot be a theorem. Hence $F^{\ast }$ is not provable in PA. Moreover its negation is equivalent to PA's consistency statement and is unprovable by the second incompleteness theorem. Thus this fixed point is, under the stated hypothesis, undecidable rather than an unboxed reflection principle.

Write $\square A$ for the arithmetical provability predicate applied to the Gödel number of $A$. Use the Hilbert-Bernays-Löb derivability conditions that belong to the course meaning of a provability predicate: external necessitation rule, internal modus ponens rule, and positive introspection. No incompleteness theorem is needed here.

If $S⊢G$, necessitation rule gives $S⊢\square G$. The displayed Gödel sentence equivalence also gives $S⊢\neg \square G$. These two theorems contradict the assumption that $S$ is a consistent first-order theory. ThereforeFor this part alone, the external necessitation rule condition suffices; the stronger internal conditions are used in the next part.

By part (c), one-consistency gives Sigma-1 soundness and hence ordinary consistency. First, $S$ cannot prove $\neg G$: if it did, the diagonal equivalence would give $S⊢\square G$. This is a ${\Sigma }_1$ sentence, so soundness would make it true. The stipulated standard interpretation of the arithmetical provability predicate would then supply an actual proof of $G$, contradicting consistency.

Now suppose $S⊢\square \neg G$. This too is a ${\Sigma }_1$ theorem. Soundness and the standard correctness of the predicate would imply that there is an actual $S$-proof of $\neg G$, which was just excluded. ThereforeThe two applications of soundness concern the provability assertions, not the truth of arbitrary arithmetic sentences.

Abbreviate the arithmetical provability predicate by $\square$, with the Hilbert-Bernays-Löb derivability conditions understood. Löb's theorem saysChoose a diagonal sentence $D$ with $S⊢D\leftrightarrow (\square D\to A)$. Internalizing its forward implication gives $S⊢\square D\to \square (\square D\to A)$. Internal modus ponens rule gives $\square (\square D\to A)\to (\square \square D\to \square A)$, while positive introspection gives $\square D\to \square \square D$. Combining these yields $S⊢\square D\to \square A$.

Now use the assumed theorem $\square A\to A$. It follows that $S⊢\square D\to A$, and the reverse direction of the diagonal equivalence gives $S⊢D$. External necessitation rule gives $S⊢\square D$, and the previously derived implication gives $S⊢A$. This proves the theorem from the supplied fixed-point property and the derivability conditions.

The Gödel second incompleteness theorem states that a consistent arithmetic theory in this setting cannot prove its own consistency sentence ${\mathrm{Con}}_S=\neg \square \perp$. If it could, it would prove $\square \perp \to \perp$; applying Löb's theorem with $A=\perp$ would make it inconsistent. Therefore $\boxed{S⊬{\mathrm{Con}}_S}$ when $S$ is consistent.

Positive introspection says that a boxed assertion can itself be boxed. For an arithmetical provability predicate it is the third of the Hilbert-Bernays-Löb derivability conditions: provability of $A$ implies provability of the assertion that $A$ is provable. It follows from formalized Sigma-1 completeness when provability assertions are Sigma-1. It is not the reflection principle $\square A\to A$.

Provability logic studies modal principles valid when $\square A$ is interpreted by an arithmetical provability predicate. The distribution axiom corresponds to formalized modus ponens rule, necessitation corresponds to verification of actual proofs, and the Löb axiom is the Internal form of Löb's theorem. The standard system capturing these principles is Gödel-Löb logic.