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.
Keywords
- Schematic systems
- Unfolding
- Finitist arithmetic
- Non-finitist arithmetic
- Feasible arithmetic
- Inductive definitions
2010 Mathematics Subject Classification
To Sol, with gratitude for his intellectual inspiration and friendship
Notes
- 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.
Note that this relativization also includes axioms such as \(0\in \mathsf {N}\) and \((\forall x\in \mathsf {N})(x'\in \mathsf {N})\).
- 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.
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.
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.
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.
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.
We assume that \(\subseteq \) defines the characteristic function of the initial subword relation. Further, we employ infix notation for these binary function symbols.
- 9.
These variables are syntactically different from the \(\mathsf {FEA}\) variables \(\alpha _0,\alpha _1,\ldots \).
- 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.
We can assume that only functions built from concatenation and multiplication are permissible bounds for the recursion.
- 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.
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.
References
Beeson, M.J.: Foundations of Constructive Mathematics: Metamathematical Studies. Springer, Berlin (1985)
Buchholtz, U.: Unfolding of systems of inductive definitions. Ph.D. thesis, Stanford University (2013)
Buchholtz, U., Jäger, G., Strahm, T.: Theories of proof-theoretic strength \(\Psi (\Gamma _{\Omega +1})\). In: Probst, D., Schuster, P. (eds.) Concepts of Proof in Mathematics, Philosophy, and Computer Science of Ontos Mathematical Logic. vol. 6, pp. 115–140. De Gruyter (2016)
Clote, P.: Computation models and function algebras. In: Griffor, E. (ed.) Handbook of Computability Theory, pp. 589–681. Elsevier (1999)
Cobham, A.: The intrinsic computational difficulty of functions. In: Logic, Methodology and Philosophy of Science II, pp. 24–30. North Holland, Amsterdam (1965)
Eberhard, S.: Weak Applicative Theories, Truth, and Computational Complexity. Ph.D. thesis, Universität Bern (2013)
Eberhard, S.: A feasible theory of truth over combinatory algebra. Ann. Pure Appl. Log. 165(5), 1009–1033 (2014)
Eberhard, S., Strahm, T.: Weak theories of truth and explicit mathematics. In: Berger, U., Diener, H., Schuster, P., Seisenberger, M. (eds.) Logic, Construction, Computation, pp. 157–184. Ontos Verlag (2012)
Eberhard, S., Strahm, T.: Unfolding feasible arithmetic and weak truth. In: Achourioti, D., Galinon, H., Fujimoto, K., Martinez, J. (eds.) Unifying the Philosophy of Truth, pp. 153–167. Springer (2015)
Feferman, S.: Transfinite recursive progressions of axiomatic theories. J. Symb. Log. 27, 259–316 (1962)
Feferman, S.: Systems of predicative analysis. J. Symb. Log. 29(1), 1–30 (1964)
Feferman, S.: A language and axioms for explicit mathematics. In: Crossley, J. (ed.) Algebra and Logic of Lecture Notes in Mathematics, vol. 450, pp. 87–139. Springer (1975)
Feferman, S.: A more perspicuous system for predicativity. In: Konstruktionen vs. Positionen I. de Gruyter, pp. 87–139. Berlin (1979)
Feferman, S.: Reflecting on incompleteness. J. Symb. Log. 56(1), 1–49 (1991)
Feferman, S.: Gödel’s program for new axioms: Why, where, how and what? In: Hájek, P. (ed.) Gödel 1996 of Lecture Notes in Logic, vol. 6, pp. 3–22. Springer, Berlin (1996)
Feferman, S.: Operational set theory and small large cardinals. Inf. Comput. 207, 971–979 (2009)
Feferman, S.: Turing’s Thesis: Ordinal logics and oracle computability. In: Cooper, S.B., Van Leeuwen, J. (eds.) Alan Turing: His Work and Impact, pp. 145–150. Elsevier (2013)
Feferman, S.: The operational perspective: three routes. In: Kahle, R., Strahm, T., Studer, T. (eds.) Advances in Proof Theory of Progress in Computer Science and Applied Logic, vol. 28, pp. 269–289. Birkhäuser (2016)
Feferman, S., Spector, C.: Incompleteness along paths in progressions of theories. J. Symb. Log. 27, 383–390 (1962)
Feferman, S., Strahm, T.: The unfolding of non-finitist arithmetic. Ann. Pure Appl. Log. 104(1–3), 75–96 (2000)
Feferman, S., Strahm, T.: Unfolding finitist arithmetic. Rev. Symb. Log. 3(4), 665–689 (2010)
Ferreira, F.: Polynomial time computable arithmetic. In: Sieg, W. (ed.) Logic and Computation, Proceedings of a Workshop held at Carnegie Mellon University, 1987, Contemporary Mathematics, vol. 106, pp. 137–156. American Mathematical Society, Providence, Rhode Island (1990)
Franzén, T.: Transfinite progressions: a second look at completeness. Bull. Symb. Log. 10(3), 367–389 (2004)
Gödel, K.: Collected Works. In: Feferman, S. et al., (eds.) Vol. II. Oxford University Press, New York (1990)
Jäger, G.: On Feferman’s operational set theory OST. Ann. Pure Appl. Log. 150(1–3), 19–39 (2007)
Kreisel, G.: Ordinal logics and the characterization of informal concepts of proof. In: Proceedings International Congress of Mathematicians, 14–21 August 1958, pp. 289–299. Cambridge University Press, Cambridge (1960)
Kreisel, G.: Mathematical logic. In: Saaty, T.L. (ed.) Lectures on Modern Mathematics, vol. 3, pp. 95–195. Wiley, New York (1965)
Kreisel, G.: Principles of proof and ordinals implicit in given concepts. In: Kino, A., Myhill, J., Vesley, R.E. (eds.) Intuitionism and Proof Theory, pp. 489–516. North Holland, Amsterdam (1970)
Kripke, S.: Outline of a theory of truth. J. Philos. 72(19), 690–716 (1975)
Rathjen, M.: The role of parameters in bar rule and bar induction. J. Symb. Log. 56(2), 715–730 (1991)
Schütte, K.: Eine Grenze für die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik. Archiv für Mathematische Logik und Grundlagen der Mathematik 7, 45–60 (1964)
Sieg, W.: Herbrand analyses. Arch. Math. Log. 30(5+6), 409–441 (1991)
Strahm, T.: The non-constructive \(\mu \) operator, fixed point theories with ordinals, and the bar rule. Ann. Pure Appl. Log. 104(1–3), 305–324 (2000)
Tait, W.: Nested recursion. Mathematische Annalen 143, 236–250 (1961)
Tait, W.: Finitism. J. Philos. 78, 524–546 (1981)
Turing, A.: Systems of logic based on ordinals. Proc. London Math. Soc. 2nd Ser. 45(I), 161–228 (1939)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-63334-3_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-63332-9
Online ISBN: 978-3-319-63334-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)