Abstract
Gentzen’s 1936 consistency proof for first-order arithmetic (Gentzen, Math Ann, 112:493–565, 1936), located between the first (Gentzen, Archiv für mathematische Logik und Grundlagenforschung, 16:97–118, 1974) and third proofs (Gentzen, Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, 4:19–44, 1938), was a work during his “transition period 1936–1938.” (Hereafter, we call the first proof the 1935 proof, the second the 1936 proof and the third the 1938 proof.) On the one hand, the 1936 proof inherited from the 1935 proof the method of “finitist (finit)” interpretation for first-order arithmetical formulas.
We would like to express our deep gratitude to Wilfried Sieg for his valuable comments on our work. In particular, we had fruitful discussions when the first author invited him to Waseda University in 2015. In addition, special thanks is due to the anonymous referees for careful readings, many helpful comments, and suggestions. The first author is partially supported by KAKENHI (Grant-in-Aid for Young Scientists (B)) 16K16690. The second author is supported by KAKENHI (Grant-in-Aid for JSPS Fellows) 16J04925.
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- 1.
[Arai 2002, p. 438].
- 2.
- 3.
Here, we should explain the literatures in more detail. Historically speaking, the first reconstruction of the 1935 proof was by Bernays (1970) and this proof was further discussed by Kreisel in (1971). More recently, this proof was discussed by von Plato in (2009). As to the connection between the 1935 and 1936 proofs, the paper Negri (1980) by Maurizio Negri should be mentioned here. He reconstructed the 1935 proof using ordinals up to \(\varepsilon _{0}\), which would create a close relationship between the 1935 and 1936 proofs. (A similar analysis was given in Sundholm’s Ph.D. thesis Sundholm 1983.) In these papers, authors focused only on mathematical sides of Gentzen’s proofs, while we focus also on a conceptual side of the 1936 proof. Moreover, as we will see in Sect. 5.2, we will investigate a uniform idea behind Gentzen’s three consistency proofs.
The content of this paper differs from our previous work (Akiyoshi and Takahashi 2013) written in Japanese with respect to the three main points. First, while the aim of Akiyoshi and Takahashi (2013) was to give only a uniform interpretation of Gentzen’s proofs, we also considered questions naturally arising from Sieg’s paper Sieg (2012). Secondly, we use another version of normalization trees, namely, a version of normalization trees reformulated with (possibly) non-well-founded trees. This notion makes it easier to see a connection between the 1936 proof and some notions of intuitionism like spreads and choice sequences. Finally, Sect. 5.3 of the present paper gives another proof of Kreisel’s no-counterexample interpretation, using normalization trees.
- 4.
[Sieg 2012, p. 123].
- 5.
The theory includes the principle of tertium non datur. Hilbert’s aim in this paper was to justify the use of this principle by means of a consistency proof.
- 6.
In fact, Sieg has explained background of not only the 1936 proof but also the 1935 proof. In the present paper, we concentrate on an analysis of the 1936 proof in the light of Sieg’s explanation.
- 7.
As to Hilbert’s exposition of the epsilon substitution method, see Hilbert (1928).
- 8.
- 9.
Several studies have already pointed out the relationship between Gentzen’s 1935 and 1936 proofs and the no-counterexample interpretation. Cf. Kreisel (1971), Sieg and Parsons (1938), Tait (2005). Furthermore, according to Sieg and Parsons (1938), Tait (2001), the idea of the no-counterexample interpretation is found in Gödel’s notes for his lecture in 1938.
- 10.
In the notation of Hilbert (1931), (TND) is \((x)\mathfrak {A}(x) \vee (Ex)\bar{\mathfrak {A}}(x)\).
- 11.
We owe this outline to Sieg. Cf. [Sieg 2012, pp. 102–105].
- 12.
This English translation is by Sieg.
- 13.
[Sieg 2012, p. 92, p. 103].
- 14.
Cf. [Sieg 2012, p. 107]. Sieg’s explanation can be paraphrased as follows: To apply the rule (\(HR^*\)), one needs to show that \(\mathfrak {A(z)}\) is correct for an arbitrary numeral \(\mathfrak {z}\). Because of Hilbert’s finitist attitude, this must be shown by giving an effective method to verify the correctness of \(\mathfrak {A}(0)\), \(\mathfrak {A}(1)\), \(\mathfrak {A}(2)\) and so on. The concept of an infinite sequence of natural numbers is used here.
- 15.
[Sieg 2012, p. 114].
- 16.
The distinction of purely formal correctness proofs from formal correctness proofs does not matter to our present concern; neither does the distinction of semi-contentual correctness proofs from contentual correctness proofs. Thus we do not consider these distinctions. In the both cases, the former is subsumed into the latter.
- 17.
[Sieg 2012, p. 117].
- 18.
For Gentzen, the statability of a reduction procedure for \(\Gamma \) gives a meaning to \(\Gamma \) from his “finitist” standpoint. It is not easy to estimate the exact strength of Gentzen’s finitist standpoint. As far as we know, his standpoint should be constructive in the following sense. First, all infinite totalities must be generated by some finitary rules (Gentzen 1936, pp. 524–525, 1969, p. 162). For example, the totality of all natural numbers is generated from 0 by the successor rule. Second, one must avoid the use of the principle of the excluded middle for non-decidable predicates (Gentzen 1936, pp. 527–528, 1969, pp. 164–165).
- 19.
- 20.
Cf. [Takeuti 1987, Definition 12.2.].
- 21.
In fact, for the claim that the 1936 proof is both contentual and formal, it is not necessary to show that the main lemma of the 1936 has these consequences. It suffices to show simply that the 1936 proof has both of the features of contentual correctness proofs and formal correctness proofs. Our strategy for the argument in this section comes from the aim of the next section.
- 22.
To formulate reduction steps of the 1936 proof by means of finite notations for infinitary derivations, we need to insert sufficiently many \(\mathsf {E}\)-rules into a given \(\mathsf {Z}\)-derivation h. Moreover, we also need the function \(\phi \) to delete the inserted \(\mathsf {E}\)-rules, since, of course, the proof system Gentzen used to provide the 1936 proof does not include the \(\mathsf {E}\)-rule.
- 23.
Strictly speaking, the above step is a special case of the reduction step that Gentzen formulated in [Gentzen 1936, Sect. 14.25], since he used not the cut-rule such as \(\mathsf {R}_C\) but the “chain-rule (Kettenschluss),” one of whose instances is the cut-rule. Recently, Buchholz analyzed the chain-rule by means of finite notations for infinitary derivations (Buchholz 2015).
- 24.
As to monotone bar induction, see [Troelstra and van Dalen 1988, Chap. 4, Sect. 8].
- 25.
One can read off this idea in Gentzen’s explanation for reduction steps of the 1936 steps. See [Gentzen 1936, Sect. 14.2].
- 26.
In the future works, we will investigate the exact strength of Gentzen’s finitist standpoint and examine whether our argument can be formalized in this standpoint.
- 27.
Here, we slightly changed Buchholz’s actual observation. The actual observation is that for a supposed \(\mathsf {Z}^*_0\)-derivation h of the empty sequent \(\emptyset \), the step \(\phi (h) \mapsto \phi (h[0])\) is a main reduction step of the 1938 proof.
- 28.
For the operational reductions, see [Gentzen 1938, Sect. 3.5].
- 29.
- 30.
Cf. Buchholz (1997). Buchholz defined also the function \(\mathsf {hgt}^* (\mathfrak {a})\) of a nominal form \(\mathfrak {a}\), to describe the ordinal assignment in the 1938 proof by means of finite notations for infinitary derivations. Here, we do not need to describe it, so we omit the definition of this function.
- 31.
[Gentzen 1936, pp. 523–524].
- 32.
[Gentzen 1936, p. 526].
- 33.
As to the definition of a boundary inference, we refer to Takeuti’s definition [Takeuti 1987, p. 79].
- 34.
In the future, when we will have more historical resources, we will investigate this point in more detail.
- 35.
The difference is our use of finite notations for infinitary derivations instead of heavy coding.
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Akiyoshi, R., Takahashi, Y. (2017). Contentual and Formal Aspects of Gentzen’s Consistency Proofs. In: Yang, SM., Lee, K., Ono, H. (eds) Philosophical Logic: Current Trends in Asia. Logic in Asia: Studia Logica Library. Springer, Singapore. https://doi.org/10.1007/978-981-10-6355-8_6
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