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Unfolding Schematic Systems

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Feferman on Foundations

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 13))

Abstract

The notion of unfolding a schematic formal system was introduced by Feferman in 1996 in order to answer the following question: Given a schematic system \(\mathsf {S}\), which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted \(\mathsf {S}\)? After a short summary of precursors of the unfolding program, we survey the unfolding procedure and discuss the main results obtained for various schematic systems S, including non-finitist arithmetic, finitist arithmetic, feasible arithmetic, and theories of inductive definitions.

To Sol, with gratitude for his intellectual inspiration and friendship

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Notes

  1. 1.

    The constants \(\mathsf {sc}\) and \(\mathsf {pd}\) as well as the relation symbol \(\mathsf {N}\) are used instead of the symbols \(\mathsf {Sc}^{\star }\), \(\mathsf {Pd}^{\star }\), and \(\mathsf {U}_{\mathsf {NFA}}\) mentioned in the informal description above.

  2. 2.

    Note that this relativization also includes axioms such as \(0\in \mathsf {N}\) and \((\forall x\in \mathsf {N})(x'\in \mathsf {N})\).

  3. 3.

    Observe that \(\mathsf {nat}\) is alternatively definable from the remaining predicate axioms by \(x\in \mathsf {nat}\leftrightarrow (\exists y\in \mathsf {N})(x=y)\).

  4. 4.

    Observe that derivability of rules is a dynamic process as we unfold \(\mathsf {FA}\). In particular, new rules of inference obtained by \((\mathsf {Subst'})\) allow us to establish new derivable rules, to which in turn we can apply \((\mathsf {Subst'})\). In particular, the usual rule of induction

    $$ \dfrac{\Gamma \,\rightarrow \,A[0] \quad \Gamma ,\, u\in \mathsf {N},\, A[u]\,\rightarrow \,A[u']}{\Gamma ,\, v\in \mathsf {N}\,\rightarrow \,A[v]} $$

    is an immediate consequence of \((\mathsf {Subst'})\) applied to rule (3) of \(\mathsf {FA}\). Moreover, the substitution rule in its usual form as stated in Sect. 2,

    $$\begin{aligned} \dfrac{\Sigma [\bar{P}]}{ \Sigma [\bar{B}/\bar{P}]} {(\mathsf {Subst})} \end{aligned}$$

    is readily seen to be an admissible rule of inference of \(\mathcal {U}_0(\mathsf {FA})\).

  5. 5.

    In the formulation of the rules below we use a binary relation \(\prec \) whose characteristic function is given by a closed term \(t_{\prec }\) for which \(\mathcal {U}_0(\mathsf {FA})\) proves \(t_{\prec }: \mathsf {N}^2 \rightarrow \{0,1\}\). We write \(x \prec y\) instead of \(t_{\prec }xy=0\) and further assume that \(\prec \) is a linear ordering with least element 0, provably in \(\mathcal {U}_0(\mathsf {FA})\).

  6. 6.

    In the formulation of this rule, we have used the shorthand \(r \prec x \supset A\) for the formula \(t_{\prec } r x =1 \vee A\).

  7. 7.

    Given two words \(w_1\) and \(w_2\), the word \(w_1 \boxtimes w_2\) denotes the length of \(w_2\) fold concatenation of \(w_1\) with itself.

  8. 8.

    We assume that \(\subseteq \) defines the characteristic function of the initial subword relation. Further, we employ infix notation for these binary function symbols.

  9. 9.

    These variables are syntactically different from the \(\mathsf {FEA}\) variables \(\alpha _0,\alpha _1,\ldots \).

  10. 10.

    It is important to note that we do not have a predicate \(\mathsf {W}\) for binary words in our language, since this would allow us to introduce (hidden) unbounded existential quantifiers via formulas of the form \(\mathsf {W}(t)\). Thus it is necessary to have two separate sets of variables for words and operations, respectively.

  11. 11.

    We can assume that only functions built from concatenation and multiplication are permissible bounds for the recursion.

  12. 12.

    Recall that by expanding the definition of the \(\le \) relation, the formula \(A[\beta ]\) stands for the assertion \((\exists \gamma \le \tau [\bar{\alpha },\beta ])(t_F(\bar{\alpha },\beta ) =\gamma )\).

  13. 13.

    Recall that in Feferman’s original definition of unfolding in [15], a truth predicate is used in order to describe the full unfolding of a schematic system.

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  1. Institut für Informatik, Universität Bern, Neubrückstrasse 10, CH-3012, Bern, Switzerland

    Thomas Strahm

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  1. Institut für Informatik, Universität Bern, Bern, Switzerland

    Gerhard Jäger

  2. Department of Philosophy, Carnegie Mellon University Dept. Philosophy, Pittsburgh, Pennsylvania, USA

    Wilfried Sieg

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Strahm, T. (2017). Unfolding Schematic Systems. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_8

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