Abstract
I provide an interpretation of Wittgenstein’s much criticised remarks on Gödel’s First Incompleteness Theorem in a paraconsistent framework: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was consequent upon his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the model-theoretic features of paraconsistent arithmetics match with many intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.
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Notes
- 1.
In the correspondence between the two mathematicians, as Dawson points out, Zermelo “failed utterly to appreciate Gödel’s distinctions between syntax and semantics” (Dawson 1984, p. 80).
- 2.
Wittgenstein’s remarks had as their background system the one of Russell and Whitehead’s Principia mathematica (with slight modifications). However, this is a minor point, and sticking to PA allows us to follow a standard way of presenting Gödel’s Theorems.
- 3.
Or, at least, these are our bona fide intuitions when we formulate the theory. The existence of non-standard models shows that things are not so straightforward. I will come to this in a subsequent note.
- 4.
Th is not recursive, but semirecursive (see Gödel 1931, p. 606); however, this is of no importance here.
- 5.
One of the consequences of Gödel’s First Theorem is that (first-order) PA is not, as model theorists say, categorical. This means that from Gödel’s results follows the existence of non-standard models of PA, structurally different from \(\mathbb{N}\). In particular, there is no way to constrain the variables of the theory so that they range exclusively on ordinary natural numbers. In 1957, Goodstein had already claimed that “Wittgenstein with remarkable insight said in the early thirties that Gödel’s results showed that the notion of a finite cardinal could not be expressed in an axiomatic system and that formal number variables must necessarily take values other than natural numbers” (Goodstein 1957, p. 551). More recently, Floyd and Putnam have credited the “notorious paragraph” 8 of the Appendix 1 to Part I of Wittgenstein’s Bemerkungen with a “philosophical claim of great interest” precisely on the role of non-standard models and ω-inconsistency. The claim is to the effect that “if one assumes (and, a fortiori, if one actually finds out) that ¬P [where P is assumed to be the Gödel sentence of the relevant system] is provable in Russell’s system one should (or, as Wittgenstein actually writes, one ‘will now presumably’) give up the ‘translation’ of P by the English sentence ‘P is not provable”’ (Floyd and Putnam 2000, p. 625). The point is that if a theory proves ¬P (which may be obtained simply by adding it as an axiom), then it is ω-inconsistent, but consistent. Being consistent, it is supposed to have a model. However, being ω-inconsistent, its model has to be structurally different from the standard model of arithmetics, \(\mathbb{N}\). It is a non-standard model, and the “translation” of P as “P is not provable” becomes untenable in this context.
- 6.
An anonymous referee has appropriately pointed out to me.
- 7.
In Kleene’s words: “Gödel’s sentence ’I am unprovable’ is not paradoxical. We escape paradox because (whatever Hilbert may have hoped) there is no a priori reason why every true sentence must be provable […]. The sentence A p (p), which says ’I am unprovable’, is simply unprovable and true” (Kleene 1976, p. 54).
- 8.
- 9.
In particular, Priest may disagree with the picture of Wittgenstein’s attitude towards Gödel proposed here (see Priest 2004).
- 10.
For a quick review, see Bremer (2005, Chap. 13).
- 11.
- 12.
- 13.
- 14.
One of the consequences of the downward Theorem is the so-called Skolem paradox. Since set theory can be expressed in a first-order language, it has a model whose domain has the cardinality of the set of natural numbers. However, within set theory we can prove the existence of sets whose cardinality is more than denumerable.
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Acknowledgements
The non-technical parts of this work draw on a paper published in Philosophia Mathematica, 17: 208–219, with the title “The Gödel Paradox and Wittgenstein’s Reasons”. I am grateful to Oxford University Press and to the Editors of Philosophia Mathematica for permission to reuse that material. I am also grateful to an anonymous referee for helpful comments on this expanded version.
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Berto, F. (2013). Wittgenstein on Incompleteness Makes Paraconsistent Sense. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_14
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