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Gödel Blooming: the Incompleteness Theorems from a Paraconsistent Perspective

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Gödel Blooming: the Incompleteness Theorems from a Paraconsistent Perspective Gödel blooming: the Incompleteness Theorems from a paraconsistent perspective Walter Carnielli Centre for Logic, Epistemology and the History of Science and Department of Philosophy University of Campinas - Unicamp, Campinas, SP, Brazil David Fuenmayor Department of Mathematics and Computer Science Freie Universität Berlin e-mail: [email protected], [email protected] Abstract This paper explores the general question of the validity of Gödel’s incompleteness theorems by examining the respective arguments from a paraconsistent perspective, while employing combinations of modal logics with Logics of Formal Inconsistency (LFIs). For this purpose, abstract versions of the incompleteness theorems, employing provabil- ity logic, need to be carefully crafted. This analysis considers distinct variants of the notion of consistency for formal systems, which, to- gether with the lighter character of the negation operator of the LFIs, enable new formalization variants of the Gödelian arguments, eventu- ally leading to some thought-provoking conclusions. We show that the standard formulation of Gödel’s theorems is not valid under some weak LFIs: a valid reconstruction requires further premises corresponding to the consistency (in the sense of LFIs) of particular formulas. This readily leads us to a reformulation of Gödel’s theorems as an exis- tence claim. In this paper we also aim at showcasing the convenience of working with modern proof assistants (in this case Isabelle/HOL), which enable much faster and accurate feedback on verifying or falsi- fying hypotheses during the process of formal proof reconstruction. Keywords: Gödel’s incompleteness theorems; Provability Logic; paracon- sistency; Logics of Formal Inconsistency; Isabelle/HOL. 1 1 How universal are Gödel’s arguments? In a rough and intuitive formulation, Gödel’s first incompleteness theorem (G1) says that, for certain consistent formal systems, there are (true) sen- tences that they cannot decide, i.e., neither prove nor disprove; the second incompleteness theorem (G2) says that such a system cannot prove its own consistency. A formidable amount of papers deals with explanations or in- terpretations of the Gödelian arguments, but few of them touch on the limits of the Gödelian objection. Gödel formulated his incompleteness theorems1 employing notions such as axiomatic systems, primitive recursive arithmetic, arithmetization/numbering, representability/interpretability, consistency, completeness, diagonalization, etc; already a quick literature survey will reveal several different (and in some cases non-equivalent) ‘formalizations’, or more appropriately, explications, of these notions. In this paper we focus on the notions of consistency and negation from the point of view of paraconsistent logic. We aim at examin- ing different, alternative notions of consistency employing both classical and paraconsistent negation, and then investigating their role in Gödel’s argu- ments with the help of the proof assistant Isabelle/HOL (henceforth Isabelle) [NPW02]. For this sake, we will abstract away the complexities of Gödel’s arithmetization procedure and assume the corresponding fixed-point (diag- onalization) lemma as a premise. We will employ for this (a paraconsistent version of) provability logic [Ver17; Boo95]. It is important to note that we do not aim at a thorough formal recon- struction of Gödel’s proofs (including the arithmetization procedure) using a proof assistant, as previously done by Paulson [Pau15] in Isabelle and O’Connor in Coq [OCo05]. Special mention deserves the work of John Har- rison, who provides a formal reconstruction of G2 within his proof assistant HOL-Light by employing provability logic in a similar spirit as ours (cf. his related work [Har09, Ch. 7]). With our formal reconstruction work in Isabelle (Section 6), we aim rather at introducing a framework for experimenting, in the context of Gödel’s proofs, with different notions of consistency, and from a paraconsistent per- spective. We employ modal logic, both as a paraconsistent and as a provabil- ity logic, drawing upon the shallow semantical embeddings (SSE) approach [Ben19]. SSE rests on the adoption of Church’s simple type theory (STT) [BA19] as an expressive higher-order meta-language into which the logical 1There is some controversy in referring to Gödel’s results as either ‘a theorem’ or ‘theorems’. Saul Kripke (in private conversation with the first author) insists that we should refer jointly to both as ‘Gödel’s theorem’, since G2 is a corollary of G1. We prefer to maintain the plural. 2 connectives of (a combination of) target logics can be ‘translated’ or ‘embed- ded’, in such a way that the target logic becomes a fragment of STT.2 In doing this, we can employ the proof assistant Isabelle, whose logic conserva- tively extends STT, to reconstruct and assess different formalized variants of Gödel’s arguments. We make the Isabelle source files for this formalization work freely available [CF20] and encourage the interested reader to carry out further experiments (e.g. by varying the encoded logic or the notions of consistency), and thus to further expand and improve on this work. In view of the abundant literature and approaches towards Gödel’s re- sults, we are obliged to restrict ourselves to considering only a few sources. Succinct, but self-contained and fairly detailed, discussions emphasizing most important points in Gödel’s proofs can be found e.g. in the works of Smoryński [Smo77, Sec. 1–2], Epstein & Carnielli [EC08, Ch. 23–24], and in the corre- sponding article in the Stanford Encyclopedia of Philosophy [Raa18]. We will draw upon them as sources for our formal reconstruction work in Section 6. Section 2 raises the question about the range of Gödel’s theorems, setting the stage for an analysis of the dependence of the Gödelian arguments on standard logical conventions. Section 3 introduces the paraconsistentist pro- gram, and foresees some difficulties as regards the validity of the Gödelian objection in more subtle logical scenarios. Section 4 discusses the Logics of Formal Inconsistency, and justifies the choice of the logic RmbC among eli- gible paraconsistent scenarios. Section 5 illustrates the mechanism by which we add a provability operator to RmbC, thus obtaining the logic RmbC⊕K, and analyzes several notions of consistency that emerge from this move. Sec- tion 6 is dedicated to the formal reconstruction of the Gödelian arguments with the help of Isabelle. This task requires a combination of modal and paraconsistent logics embodied in RmbC⊕K. Since such logics are not cur- rently supported off-the-shelf by mainstream proof assistants, we discuss the changes that turn Isabelle into a flexible, high-level modal and paraconsistent logic reasoner. This section also contains the most relevant technical results. Finally, Section 7 offers the main conclusions of this paper. 2The SSE technique has been employed successfully in the logical analysis of argu- mentative discourse (e.g. in the formal reconstruction of another, more metaphysical Gödelian argument [BW16; FB17], or in formal ethics [FB19b]), where it has also inspired a computer-supported approach, computational hermeneutics [FB19a], which, adhering to the slogan: ‘every formalization is an interpretation’, aims at rendering explicit the tacit conceptualizations implicit in argumentative practices. SSE also has applications in AI and normative systems [BPT20], as it supports the reuse of existing reasoning infrastruc- ture for first-order and higher-order logics for seamlessly combining and reasoning with different quantified classical and non-classical logics (including modal, deontic, epistemic and paraconsistent), many of which are well suited for normative reasoning applications. 3 2 On the range of Gödel’s theorems Gödel’s incompleteness theorems do not apply unrestrictedly to every math- ematical system. G1 does not apply for instance to Euclidean geometry. Tarski proved in 1948 [Tar98] that the first-order theory of Euclidean ge- ometry is complete and decidable,3 in the sense that every statement in its language is either a theorem (i.e., provable) or its negation is a theorem, and there is an algorithm to determine which is the case. Even non-Euclidean geometry falls outside the Gödelian barrier: if plane Euclidean geometry is a consistent theory, then so is plane hyperbolic geometry. Nor do Gödel’s theorems apply to systems of arithmetic with the addition operation only, such as Presburger arithmetic. It is necessary to have a certain critical mass of mathematical strength to be attacked by the Gödelian objection, or, in other words: to qualify as what Gödel himself defined as ‘a formal system’. Our aim in this paper is to give some first steps towards the analysis, using non-classical logics, of the applicability of Gödel’s theorems to formal systems, while exploiting the computing power of modern proof assistants. Very little investigation, if any, touches the validity of Gödel’s proofs in non- classical environments. For instance, [BS14] claims to be studying versions of Gödel’s arguments under the umbrella of non-classical logics, but just con- centrates on substructural logics: the authors show that the Gödelian rea- soning presupposes a certain amount of contraction in the underlying logic, that is, the validity of the meta-rule ‘Γ; ; ` ' implies Γ; ` '’. They then exhibit a modal system without contraction that invalidates Gödel’s argument. On the other hand, several authors including Kreisel, Feferman, Löb,
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