Abstract
By a semi-formal system we understand a proof system which includes infinitary inference rules. The use of inference rules with infinitely many premises was already suggested by David Hilbert in his paper “Die Grundlegung der elementaren Zahlentheorie” [6] and was later systematically used by Kurt Schütte in his work on proof theory. The heigths of proof trees in a semi-formal system are canonically measured by ordinals. Therefore, in contrast to Gentzen’s original approach, ordinals enter proof theoretic research via semi-formal systems in a completely canonical way.
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- 1.
Observe, however, that atomic formulas may well belong to a type. Cf., e.g., Definition 2.10 below.
- 2.
That is, languages that comprise a complete set of Boolean operations, first order variables and the first order quantifiers \(\forall \) and \(\exists \) (either explicitly or by definition) together with a set of non-logical symbols and—possibly—a set of free predicate variables.
- 3.
This ensures that the semantics defined in Definition 2.4 coincides with the usual semantics for first order languages.
- 4.
We write \(F \simeq \bigwedge \mbox{ $\left \langle { G_{\iota }\vert \,\iota \in I}\right \rangle $}\) to indicate that F is in \({\bigwedge-\mathsf{type}}\) and \(\mbox{ CS}(F) = \mbox{ $\left \langle { G_{\iota }\vert \,\iota \in I}\right \rangle $}\). Similarly we use \(F \simeq \bigvee \mbox{ $\left \langle { G_{\iota }\vert \,\iota \in I}\right \rangle $}\).
- 5.
It only says that every \(\mathfrak{M}\)-assignment \(\Phi \) that satisfies all formulas in T also satisfies F in \(\mathfrak{M}\).
- 6.
Which means that there are many term-models according to the interpretation of the relation constants.
- 7.
Cf. [1, III, Proposition 1.4].
- 8.
For full generality we need a domain \(\mathsf{S}^{k} \subseteq \mbox{ Pow}\big(\vert \mathfrak{M}\vert \big)^{k}\) for every quantifier on k-ary predicate variables. The restriction to unary predicate variables is in fact only a matter of simplifying notations.
- 9.
This is of course not surprising because weak first order logic is in fact nothing but a two sorted first order logic.
- 10.
\(\mbox{ rnk}(t \in \mathsf{M}) = \mbox{ rnk}(t\notin \mathsf{M}) = 1\) will work in most cases, especially in the case that the basis language is the first order language of \(\mathfrak{M}\).
- 11.
According to this extension M and \(\mathsf{M}^{c}\) are interpreted by the definition of CS(t ∈ M) and CS(t ∉ M), respectively.
- 12.
Observe the peculiarity that for \(m \in \vert \mathfrak{M}\vert \) we have a constant \(\underline{m}\) in \(\mathbf{L}_{\mathfrak{M}}^{+}\) and thus a constant \(\underline{\underline{m}}\) in \(\mathbf{L}_{\mathfrak{T}}^{\mathfrak{M}}\). The interpretation of \(\underline{\underline{m}}\) in the extended model \((\mathfrak{T}_{\mathbf{L}_{\mathfrak{M}}})_{\mathfrak{T}_{\mathbf{L}_{ \mathfrak{M}}}}\) is \(\underline{m}\) and the interpretation of \(\underline{m}\) in \(\mathfrak{T}_{\mathbf{L}_{\mathfrak{M}}}\) is m. Thus \((R\underline{\underline{m}}_{1},\ldots,\underline{\underline{m}}_{k})\) belongs to the diagram of \((\mathfrak{T}_{\mathbf{L}_{\mathfrak{M}}})_{\mathfrak{T}_{\mathbf{L}_{ \mathfrak{M}}}}\) iff \((R\underline{m}_{1},\ldots,\underline{m}_{k})\) belongs to the diagram of \(\mathfrak{T}_{\mathfrak{M}}\) iff \((Rm_{1},\ldots,m_{k})\) belongs to the diagram of \(\mathfrak{M}\).
- 13.
Cf. [1, II Sect. 5].
- 14.
An L-formula is X positive if the dual variable \(X^{c}\) does not occur in the Tait version of L.
- 15.
Cf. [10, Sect. 11.9] for the definition of a semi-decorated language for the constructible hierarchy L α .
- 16.
Cf. [1] for the use of extended first order- and extended \(\Pi _{1}^{1}\)-formulas.
- 17.
Since we are working with the first order logic \(\mathsf{L}(\mathfrak{M})\) of \(\mathfrak{M}\) as basis language we denote from now on truth complexity briefly by \(\mbox{ tc}_{\mathfrak{M}}(F)\) instead of \(\mbox{ tc}_{\mathsf{L}(\mathfrak{M})_{\mathfrak{M}}}(F)\).
- 18.
Cf. Remark 5.23.
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Pohlers, W. (2015). Semi-Formal Calculi and Their Applications. In: Kahle, R., Rathjen, M. (eds) Gentzen's Centenary. Springer, Cham. https://doi.org/10.1007/978-3-319-10103-3_13
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