These two types of developments are complementary, and they both contribute to formally taming the complex ramifications of the incompleteness theorems.
9.2 Connection to Paulson’s Results
The above instances are heavily based on the lemmas proved by Paulson in his Isabelle/HOL formalization of
\(\mathcal {IT}_ {1}\) (covering both the proof-theoretic and the semantic aspect) and
\(\mathcal {IT}_{2}\) [
39,
40]. Paulson formalized quite faithfully Świerczkowski’s detailed account [
61], but he also strengthened and slightly corrected it. Świerczkowski’s work applies to HF set theory [
61,
62], a classical FOL theory axiomatizing hereditarily finite sets by means of an induction principle stating that the universe is comprised of such sets only. Paulson extended Świerczkowski’s incompleteness to essential incompleteness with respect to any finite sound extension of HF set theory within the same FOL language.
Our Theorem
34’s point (1) is a restatement of Paulson’s formalized results: theorems
Goedel\(\_\)I and
Goedel\(\_\)II in [
40]. By contrast, point (2) is an upgrade of Paulson’s
Goedel\(\_\)II, applicable to any finite consistent, though possibly unsound theory. This stronger version is a more standard form of
\(\mathcal {IT}_{2}\), free from any model-theoretic dependencies. Paulson proved both
HBL \(_{1}\) and
HBL \(^{\Leftarrow }_{1}\) taking advantage of soundness, so to achieve the upgrade we had to discard
HBL \(^{\Leftarrow }_{1}\) and re-prove
HBL \(_{1}\) by replacing any semantic arguments with proofs within the HF calculus. We also removed all invocations of the
\(\varSigma _1\)-completeness lemma, which happened to depend on soundness due to Paulson’s choice of
\(\varSigma _1\)-sentence definition.
This instantiation process has offered us important feedback into the abstract results. A formal development such as ours is (largely) immune to reasoning errors, but not to missing out on useful pieces of generality. We experienced this firsthand with our assumptions about substitution. An
a priori natural choice was to assume representability of the numeral substitution
\({{\mathsf {Sb}}}: {{\mathsf {Fmla}}}_1 \times {{\mathsf {Num}}}\rightarrow {{\mathsf {Sen}}}\) (defined as
\({{\mathsf {Sb}}}(\varphi ,n) = \varphi (n)\)), part of which means (1)
. Instead, Paulson had proved (2)
. Unlike (1), Paulson’s (2) applies the term encoding function
\(\langle \_\rangle : {{\mathsf {Term}}}\rightarrow {{\mathsf {Num}}}\) to numerals as well (which are particular terms); and since his
\(\langle \_\rangle \) function is injective, it is far from the case that
\(\langle n\rangle = n\) for all numerals
n. Paulson’s version makes more sense than ours when building the results bottom-up: Representability should not discriminate numerals, but filter them through the encodings like other terms. However, top-down our version also made sense: It yielded the incompleteness theorems under reasonable assumptions, which do hold, by the way, for HF set theory—even though in a bottom-up development one is unlikely to prove them. We resolved this discrepancy through a common denominator: the representability of self-substitution
\({{\mathsf {S}}}: {{\mathsf {Fmla}}}_1 \rightarrow {{\mathsf {Sen}}}\) (Sect.
4.4), which made our results more general.
Paulson’s formalization has also inspired our abstract treatment of standard models (Sect.
4.6). Since Paulson proved
HBL \(^{\Leftarrow }_{1}\) and used classical logic, an obvious “port of entry” of his
\(\mathcal {IT}_{2}\) into our framework is Theorem
22, taking both
\(\vdash ^{\textsf {b}}\) and
\(\vdash \) to be Paulson’s provability relation (which is classical provability in a finite extension of HF set theory). But this theorem tells us nothing about the Gödel sentences’ truth. Delving deeper into Paulson’s development, we noted that, following Świerczkowski, he (unconventionally) completely avoided
Repr \(_{\Vdash }\) , and did not even define
\(\mathrel {\Vdash }\). This raised the question of whether
HBL \(^{\Leftarrow }_{1}\) and
Repr \(_{\Vdash }\) are somehow interchangeable in the presence of Standard Models standard models (on which Paulson relies heavily); and we found that they indeed are, under mild assumptions about truth (as we discuss in Sect.
4.6). This analysis has led to variants of our semantic
\(\mathcal {IT}_ {1}\), Theorems
18 and
20, which incidentally do not need classical logic. Although our Theorem
18 seemed like an excellent candidate to instantiate to Paulson’s semantic
\(\mathcal {IT}_ {1}\), its instantiation turned out to be difficult. All its assumptions were easy to fulfill based on what Paulson had already proved, except for
. Indeed, whereas Paulson proved that his proof-of relation is a
\(\varSigma _1\)-formula (which implies
by
\(\varSigma _1\)-completeness), he did not prove the same for its negation (which would imply
). Instead, we recovered Paulson’s
\(\mathcal {IT}_ {1}\) as an instance of our Theorem
24 (which requires classical logic).
There are two further improvements that we could perform to Paulson’s formalization, leveraging our abstract results: (1) replacing the soundness assumption from Paulson’s
\(\mathcal {IT}_ {1}\) with consistency, and (2) removing all traces of classical reasoning in the object logic to port Paulson’s
\(\mathcal {IT}_ {1}\) and
\(\mathcal {IT}_{2}\) to intuitionistic logic. For the first improvement, we must prove the aforementioned missing link between Paulson’s
\(\mathcal {IT}_ {1}\) and our Theorem
18, namely showing that
holds in Paulson’s setting; we are confident that this is true (any reasonable proof-of relation is a
\(\varDelta _1\)-formula, implying that its negation is a
\(\varSigma _1\)-formula), but the proof will be very laborious. The second improvement will have a large formal overlap with the first: To remove the uses of the unrestricted Excluded Middle axiom, we must prove that instances of this axiom hold intuitionistically for several formulas expressing decidable predicates, including many predicates that participate in the definition of Paulson’s
\({\mathsf {Pf}}\), as well as
\({\mathsf {Pf}}\) itself; and, in the presence of
, we have that
is equivalent to Excluded Middle holding for
\({\mathsf {Pf}}(n,\langle \varphi \rangle )\).
9.3 Connection to Results Mechanized in Other Provers
Shankar’s 1986 development. In pioneering work [
52,
53], Shankar proved formally the proof-theoretic version of
\(\mathcal {IT}_ {1}\) for any finite extension of the FOL theory Z2 [
10], i.e., he proved Z2’s finitary essential incompleteness. Z2 is a variation of HF set theory, the difference between the two being that the latter postulates an induction principle for all the HF sets, whereas the former singles out the natural numbers as those transitive HF sets that are totally ordered by membership and postulates induction for numbers only. The underlying object logic considered by Shankar was classical FOL enriched with definitions by the Skolemization of any proved “exists unique” sentences. He worked in Thm, an early version of the Boyer–Moore prover that eventually evolved into Nqthm [
6] and then ACL2 [
26]. This prover’s logic, i.e., the meta-logic of Shankar’s development, is a quantifier-free FOL enriched with induction and recursion principles for reasoning about total functions expressed in pure Lisp. This is significantly less expressive than HOL, and in fact close to primitive recursive arithmetic (PRA). Formally proving
\(\mathcal {IT}_ {1}\) within the constraints of this minimalistic meta-logic was an impressive achievement even by today’s standards.
Shankar’s development follows a similar structure to Cohen’s high-level informal presentation [
10, §9] (which Shankar cites). He proved that all partial recursive functions are representable in Z2, a result we will refer to as
\(\mathcal {RR}\). Besides being a central result in itself,
\(\mathcal {RR}\) is a convenient tool for proving Gödel’s theorems. Some proof developments for
\(\mathcal {IT}_ {1}\), including the Świerczkowski-Paulson one, do not prove
\(\mathcal {RR}\) in its generality, but prove the representability of needed functions only. On the other hand, the
\(\mathcal {RR}\) route is usually the one preferred in textbooks due to its elegance and generality. As Shankar observed, textbook proofs of
\(\mathcal {IT}_ {1}\) via
\(\mathcal {RR}\) often step from the meta-logic (where the usual informal mathematical discourse takes place) into a meta-meta-logic: The formula- and proof- manipulating functions needed for
\(\mathcal {IT}_ {1}\) are defined (as usual) as meta-level functions, then a meta-meta-level argument is being made that they are recursive, in order to conclude that they are representable. In a mechanization, however, such an argument must stay in the meta-logic. Shankar achieves this by formalizing a pure Lisp interpreter that is able to evaluate any recursive function when taking its description as input. His formulation of
\(\mathcal {RR}\) refers to this interpreter, stating that the interpreter’s partial-function behavior (in relational form) is representable in Z2. Each function needed in the proof of
\(\mathcal {IT}_ {1}\) is proved to be representable by first showing it to be equivalent to its interpreted version. Special care is required to have these definitions and proofs work in the meta-logic, where all functions must terminate—to that end, the interpreter takes an additional numeric argument representing the maximum allowed size of the computation. Using notations close to the ones in this paper and bypassing the indirection through the interpreter, Shankar’s proof of
\(\mathcal {IT}_ {1}\) can be summarized as follows. He defined a partial function
\({{\mathsf {THM}}}: {{\mathsf {Sen}}}\rightarrow \{0,1\}\) that, upon an input
\(\varphi \), enumerates all the possible proofs and:
-
terminates and returns 1 if a proof of \(\varphi \) is found;
-
terminates and returns 0 if a proof of \(\lnot \,\varphi \) is found.
In particular,
\({{\mathsf {THM}}}\) loops (i.e., is undefined) if neither
\(\varphi \) nor
\(\lnot \,\varphi \) is provable. Also, if both
\(\varphi \) and
\(\lnot \,\varphi \) are provable (meaning the considered extension of Z2 is inconsistent), then the output of
\({{\mathsf {THM}}}\) depends on whose proof comes first in the enumeration. But regardless of that, it holds that
\({{\mathsf {THM}}}(\varphi ) = 1\) implies
\(\vdash \varphi \), and
\({{\mathsf {THM}}}(\varphi ) = 0\) implies
\(\vdash \lnot \,\varphi \).
Let \(\psi \in {{\mathsf {Fmla}}}_1\) be the formula that represents the unary relation \(\{\varphi \in {{\mathsf {Sen}}}\mid {{\mathsf {THM}}}(\varphi \langle \varphi \rangle ) = 1\}\); this is obtained by (i) invoking \(\mathcal {RR}\) to produce a formula \(\chi \in {{\mathsf {Fmla}}}_2\) that represents the graph of the partial function \({{\mathsf {THM}}}\circ {{\mathsf {S}}}\) (where \({{\mathsf {S}}}\) is the self-substitution operator), and (ii) substituting \(\langle 1\rangle \) for \(\chi \)’s second variable. Let \(\textsf {CS}\) be the Cohen–Shankar sentence \(\lnot \,\psi \langle \lnot \,\psi \rangle \).
Now, assume that
\(\vdash \textsf {CS}\) or
\(\vdash \lnot \,\textsf {CS}\), meaning that
\({{\mathsf {THM}}}(\textsf {CS})\) terminates and returns 1 or 0. We have two cases, both of which contradict consistency:
-
If \({{\mathsf {THM}}}(\textsf {CS}) = 1\) (i.e., \(({{\mathsf {THM}}}\circ {{\mathsf {S}}}) (\lnot \, \psi ) = 1\)), then we have \(\vdash \lnot \,\psi \langle \lnot \,\psi \rangle \) by \({{\mathsf {THM}}}\)’s definition, and also \(\vdash \psi \langle \lnot \,\psi \rangle \) by \(({{\mathsf {THM}}}\circ {{\mathsf {S}}})\)’s representability.
-
If \({{\mathsf {THM}}}(\textsf {CS}) = 0\) (hence \(({{\mathsf {THM}}}\circ {{\mathsf {S}}}) (\lnot \, \psi ) \not = 1\)), then we have \(\vdash \lnot \,\lnot \,\psi \langle \lnot \,\psi \rangle \) by \({{\mathsf {THM}}}\)’s definition, and also \(\vdash \lnot \,\psi \langle \lnot \,\psi \rangle \) by \(({{\mathsf {THM}}}\circ {{\mathsf {S}}})\)’s representability.
The above proof, which is similar to Cohen’s proof sketch,
6 does not make explicit reference to
HBL \(_{1}\), although this is of course a consequence of
\(\mathcal {RR}\) via the representability of the “proof of” relation. In fact, the proof makes use of the representability of
\({{\mathsf {THM}}}\circ {{\mathsf {S}}}\), which is a variation of the representability of
\(\vdash \) (for particular sentences of the form
\(\varphi \langle \varphi \rangle \)) featuring a positive version of the Rosser twist discussed in Sect.
5, but at the meta-level: The considered relation is not just provability, but provability by a proof
p such that there is no proof
q of the formula’s negation occurring earlier in the enumeration.
The above argument is based on the diagonalization, though at the meta-level not at the object level as in Proposition
9. As Shankar remarked, the sentence
\(\textsf {CS}\) says “my negation is provable by a proof that comes in the enumeration before any proof of me”. This is true in the context of the above argument by contradiction, namely under the assumption that
\(\textsf {CS}\) is decided (either provable or unprovable). Indeed, from the definitions of
\(\psi \) and
\({{\mathsf {THM}}}\), we see that
\(\textsf {CS}\) says “it is not the case that a proof of
\(\textsf {CS}\) comes before a proof of
\(\lnot \,\textsf {CS}\)”, which, given the assumption, is equivalent to the above.
Let us refer to such sentence
\(\textsf {CS}\) as Cohen-Shankar sentences (without claiming historical accuracy about the ideas behind them, which seem to go back at least as far as Smullyan [
59]). They can alternatively be obtained by diagonalization in the object logic, namely using Proposition
9 and
Repr \(_\lnot \) to find
\(\textsf {CS}\) such that
, where < is the representation of the occurrence order in the enumeration of proofs used in
\({{\mathsf {THM}}}\)’s definition. While classically a Cohen–Shankar sentence is essentially the negation of a Rosser sentence, intuitionistically this is not the case. However,
\(\textsf {CS}\) can replace the Rosser sentence in our proof-theoretic Rosser-style
\(\mathcal {IT}_ {1}\), covered by Propositions
14,
15 and Theorem
16. Indeed, a bit of mining reveals that the proofs of these results are sufficiently general to accommodate both types of sentences. Given any
\(\varphi _1\) and
\(\varphi _2\), let us define the one-variable formula
\({\textsf {Twist}}_{\varphi _1,\varphi _2}(x)\) to be
. Note that, in the presence of
Repr \(_\lnot \) , we have that (i)
\(\vdash {\textsf {R}}\mathrel {\;\leftarrow \rightarrow \;}\lnot \, {\textsf {Twist}}_{{\textsf {R}},\lnot \,{\textsf {R}}}\) for all Rosser sentences
\({\textsf {R}}\), and (ii)
\(\vdash \textsf {CS}\mathrel {\;\leftarrow \rightarrow \;}{\textsf {Twist}}_{\lnot \,\textsf {CS},\textsf {CS}}\) for all Cohen–Shankar sentences
\(\textsf {CS}\). Based on this observation, we can amend the proofs of Propositions
14 and
15 by simply replacing
\(\lnot \,{\textsf {R}}\) with
\(\textsf {CS}\) and
\({\textsf {R}}\) with
\(\lnot \,\textsf {CS}\), and using (ii) instead of (i). Let us illustrate this on the crucial point (5) in Proposition
14’s proof, establishing that
\(\vdash \forall x. \,\lnot \,{\textsf {Twist}}_{{\textsf {R}},\lnot \,{\textsf {R}}}(x)\), which is used to infer
\(\vdash \lnot \,(\exists x.\,{\textsf {Twist}}_{{\textsf {R}},\lnot \,{\textsf {R}}}(x))\), hence (by (i))
\(\vdash {\textsf {R}}\), leading to a contradiction with the
\(\vdash \lnot \,{\textsf {R}}\) assumption. After the replacement, point (5) establishes that
\(\vdash \forall x.\, \lnot \,{\textsf {Twist}}_{\lnot \,\textsf {CS},\textsf {CS}}(x)\), which is used to infer
\(\vdash \lnot \,(\exists x. \,{\textsf {Twist}}_{\lnot \,\textsf {CS},\textsf {CS}}(x))\), hence (by (ii))
\(\vdash \lnot \,\textsf {CS}\), leading to a contradiction with the
\(\vdash \textsf {CS}\) assumption; and similarly for Proposition
15. Moreover, the proof of our semantic Rosser-style
\(\mathcal {IT}_ {1}\) (Theorems
19 and
20) can be straightforwardly adapted to show, under our abstract assumptions, that
\(\textsf {CS}\) is false (i.e.,
\(\lnot \,\textsf {CS}\) is true) in the standard model—a fact also proved by Cohen in his concrete setting. In conclusion, our abstract results can be migrated from Rosser to Cohen–Shankar sentences without requiring classical reasoning in the object logic.
O’Connor’s 2005 development. O’Connor proved formally the proof-theoretic version of
\(\mathcal {IT}_ {1}\) [
36,
37] for any self-representable extension of a classical FOL theory called NN [
23, §7.1], i.e., he proved the essential incompleteness of NN with respect to self-representable extensions. Self-representability of a FOL theory means that its set of axioms is represented by a one-variable formula in that theory. NN is a modification of Robinson arithmetic obtained by replacing the dichotomy axiom ( any element is either 0 or a successor) with three axioms regulating the behavior of an additional binary relation symbol for strict order, namely stating that (i) no element is smaller than 0, (ii) being smaller than the successor of an element implies being smaller than or equal to that element, and (iii) the order is total. NN has a similar (though not comparable) expressiveness to Robinson arithmetic. Like the latter, it is significantly less expressive than Peano arithmetic yet sufficient for
\(\mathcal {IT}_ {1}\) (but not for
\(\mathcal {IT}_{2}\)).
O’Connor worked in the Coq prover [
3], so his meta-logic is Coq’s underlying Calculus of Inductive Constructions [
38], an intuitionistic logic based on intensional type theory. Working out
\(\mathcal {IT}_ {1}\)’s theorem intuitionistically (though for a classical object logic)
7 was original, and revealed some interesting phenomena.
O’Connor’s development followed the informal presentation from Hodel’s textbook [
23], with some notable modifications discussed below. Following Hodel, and similarly to Shankar, he combined a representability theorem for a class of computable functions with proofs that all functions needed for
\(\mathcal {IT}_ {1}\) are in this class, hence are representable. However, unlike Hodel and Shankar, O’Connor did not prove representability for all recursive functions (a result we denoted by
\(\mathcal {RR}\)), but stopped at the representability of
primitive recursive functions—we will refer to this latter result as
\(\mathcal {PR}\). This restriction made the proofs that certain functions are in the considered class more difficult—notably, he reported on the difficulty of establishing that substitution is primitive recursive. On the other hand, O’Connor’s formalized representability result is stronger than Shankar’s on the theory expressiveness dimension, since it is proved for the minimalistic theory NN.
O’Connor proved a version of \(\mathcal {IT}_ {1}\) that would classically read as follows: For any consistent self-representable extension of NN, there exists a sentence \(\varphi \) such that neither \(\varphi \) nor \(\lnot \,\varphi \) is provable. Due to the intuitionistic meta-logic, O’Connor preferred the intuitionistically stronger (and classically equivalent) formulation: For any self-representable extension of NN, there exists a sentence \(\varphi \) such that, if \(\varphi \) or \(\lnot \,\varphi \) are provable, then that extension proves everything (i.e., is inconsistent). Another consequence of the intuitionistic meta-logic is the need for an additional assumption: that the given extension’s set of axioms is decidable, i.e., its (meta-level) membership predicate satisfies Excluded Middle.
The above universally quantified
\(\varphi \) is witnessed by a Rosser sentence constructed via diagonalizaton, so the result essentially falls under Propositions
9,
10 and Theorem
16, where both
\(\vdash ^{\textsf {b}}\) and
\(\vdash \) are taken to be deduction in a self-representable extension of NN. (Note that all the FOL theories of interest for
\(\mathcal {IT}_ {1}\) can already be represented in NN, not only in an extension of NN; and the corresponding (slightly weaker) version of O’Connor’s result assuming NN-representability instead of self-representability is obtained by taking
\(\vdash ^{\textsf {b}}\) to be deduction in NN and
\(\vdash \) to be deduction in the considered extension.)
Since here the FOL infrastructure is fixed, self-representability is equivalent to representability of the “proof of” relation (which O’Connor proved), hence it implies
HBL \(_{1}\) (which he did not mention explicitly but inlined in his proof). Incidentally, O’Connor’s formalization improves on Hodel’s account, who unnecessarily added an axiom to NN for coping with Rosser’s trick [
37, §6.4].
O’Connor’s self-representability assumption in
\(\mathcal {IT}_ {1}\) is more general than the standard recursive axiomatizability assumption. In informal accounts of essential incompleteness including Hodel’s, this more general result is usually inlined in the proof and only the end result is stated, which assumes not self-representability but recursive axiomatizability; an exception is the account of Feferman, who assumes a generalized form of self-representability (namely representability in a sub-theory) in his statements of
\(\mathcal {IT}_ {0.5}\) and
\(\mathcal {IT}_{2}\) (Theorems 5.3 and 5.6 in [
13]). In a formal account, such more general results are valuable for easier reusability across different instances.
O’Connor did not prove that all recursively axiomatizable extensions of NN are self-representable (which would have followed from \(\mathcal {RR}\)). However, he used his \(\mathcal {PR}\) together with a proof that Peano arithmetic has its axioms primitively recursive to instantiate \(\mathcal {IT}_ {1}\) to Peano arithmetic. He also proved the consistency of this theory (by showing that the natural numbers form a model, via a semantic interpretation function wrapped up in a negative translation to ensure classical validity within the intuitionistic meta-logic). Thus, he obtained the theory’s unconditional incompleteness.
Harrison’s 2009-2010 development. Harrison [
21] proved formally versions of
\(\mathcal {IT}_ {1}\) for theories in the language of Robinson arithmetic with
\(\le \) and < included as primitive predicate symbols. In what follows, we will refer to this language as
\(\mathcal {LA}\), and by “Robinson arithmetic” we will mean the definitional extension of Robinson arithmetic as a theory in
\(\mathcal {LA}\) (with added axioms that define
\(\le \) and <). Harrison worked in HOL Light [
20], a proof assistant belonging to the HOL family together with Isabelle/HOL and HOL4.
In his development towards
\(\mathcal {IT}_ {1}\), Harrison followed a semantic approach, based on ideas that go back to Gödel’s introduction of his original paper [
17]. The approach was promoted by Smullyan [
60] for its simplicity and elegance, and Harrison himself further elaborated and improved on it in his textbook [
19, §7]. The focus is no longer on the concept of a relation’s representability (for a given theory), but on that of a relation’s
definability in the standard model (for a given language). In our notations, definability is obtained by replacing
\(\vdash ^{\textsf {b}}\) with
\(\models \) in either the representability or the weak representabilty condition.
8 (Harrison formalized an equivalent definition of definability using valuations in the model.) The advantage of definability over representability is that the former is typically much easier to prove for concrete relations, without having to work inside a formal proof system.
\(\mathcal {LA}\) is sufficient to achieve the definability (in the standard model of natural numbers) of the relevant syntactic concepts. These include (soft) self-substitution, which gives a semantic version of diagonalization: Proposition
9 with
\(\vdash ^{\textsf {b}}\) replaced by
\(\models \). In turn, this leads to the semantic version of Tarski’s theorem on the undefinability of truth, which concludes the non-existence of a one-variable formula
\({{\mathsf {T}}}\) such that
\(\models \varphi {\mathrel {\;\leftarrow \rightarrow \;}} {{\mathsf {T}}}\langle \varphi \rangle \) for all
\(\varphi \). And after showing that provability in Robinson arithmetic
is definable, one obtains that provability is distinct from truth; in particular, for sound theories this implies the incompleteness of provability, a first version of the proof-theoretic
\(\mathcal {IT}_ {1}\). In fact, Harrison proved something more general: If a theory
T in
\(\mathcal {LA}\) is definable (in that its set of axioms is definable), then its set of provable sentences is definable, hence different from the set of true sentences. This leads to a form of essential incompleteness: Any sound definable theory in
\(\mathcal {LA}\), in particular, any extension of Robinson arithmetic with a sound definable set of axioms, is incomplete.
Harrison also pursued an alternative semantic route to
\(\mathcal {IT}_ {1}\), which does not go through Tarski’s theorem, but instead: (1) assumes (for starters) the soundness of the theory, (2) obtains a semantic version of Gödel sentences
\({\textsf {G}}\) using the semantic diagonal lemma, and (3) performs (what can be regarded as) a modification of the Gödel’s original argument (the proofs of Propositions
11 and
12), appealing to soundness whenever needed for shifting from provability to truth. The advantage of this last line of reasoning is that it can be sharpened: Noting that soundness is only needed for
\({\textsf {G}}\),
\(\lnot \,{\textsf {G}}\) and
\(\bot \), and using the fact that
\({\textsf {G}}\) is a
\(\varPi _1\)-sentence (making
\(\lnot \,{\textsf {G}}\) a
\(\varSigma _1\)-sentence) if the theory is
\(\varSigma _1\)-definable (i.e., definable by a
\(\varSigma _1\)-formula), Harrison obtained the following stronger, symmetric version of proof-theoretic
\(\mathcal {IT}_ {1}\): If a theory in
\(\mathcal {LA}\) is
\(\varSigma _1\)-definable, then (i) if it also
\(\varPi _1\)-sound then
\(\not \vdash {\textsf {G}}\) and (ii) if it also
\(\varSigma _1\)-sound then
\(\not \vdash \lnot \,{\textsf {G}}\) (where
\(\vdash \) denotes deduction from this theory, and
X-soundness or
X-completeness means soundness or completeness for all
X-sentences). And from representability and the semantic Gödel-sentence property, under the assumptions of (i), it follows that
\(\models {\textsf {G}}\). So he obtained both the proof-theoretic and the semantic component of
\(\mathcal {IT}_ {1}\).
In the above statement of
\(\mathcal {IT}_ {1}\), the
\(\varPi _1\)-soundness assumption can be replaced by consistency plus
\(\varSigma _1\)-completeness, since the latter two imply the former. Finally, using the
\(\varSigma _1\)-completeness for Robinson arithmetic (and hence for any extension), Harrison formalized an essential incompleteness generalization and strengthening of the original Gödel-style
\(\mathcal {IT}_ {1}\): For any consistent
\(\varSigma _1\)-definable extension of Robinson arithmetic, we have
\(\not \vdash {\textsf {G}}\) and
\(\models {\textsf {G}}\); and if the extension is also
\(\varSigma _1\)-sound, then
\(\not \vdash \lnot \,{\textsf {G}}\). In the presence of
\(\varSigma _1\)-completeness, the
\(\varSigma _1\)-soundness property (also called 1-consistency) is weaker than the
\(\omega \)-consistency property used originally by Gödel, which we assume in our Proposition
12 and Theorem
13.
Currently, refinements of
\(\mathcal {IT}_ {1}\) based on arithmetical hierarchy considerations are below the level of abstraction of our general framework. On the other hand, the high-level aspects of the Smullyan–Harrison semantic line of reasoning could be incorporated in this framework, which has infrastructure for both provability and truth. Our Archive of Formal Proofs entry [
44] already contains proof-theoretic and semantic versions of Tarski’s theorem.