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The provably total functions of basic arithmetic and its extensions

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Abstract

We study Basic Arithmetic, \(\textsf{BA}\) introduced by Ruitenburg (Notre Dame J Formal Logic 39:18–46, 1998). \(\textsf{BA}\) is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions of \(\textsf{BA}\) is a proper sub-class of the primitive recursive functions. Three extensions of \(\textsf{BA}\), called \(\textsf{BA}+\mathsf U\), \(\mathsf {BA_{\mathrm c}}\) and \(\textsf{EBA}\) are investigated with relation to their provably total recursive functions. It is shown that the provably total recursive functions of these three extensions of \(\textsf{BA}\) are exactly the primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem does not hold in \(\textsf{BA}\), \(\textsf{BA}+\mathsf U\), \(\mathsf {BA_{\mathrm c}}\), but holds in \(\textsf{EBA}\).

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Notes

  1. In the literature, the term “geometric logic" is usually used for a logic with infinitary disjunction, and the formulas with only finitary disjunction are called coherent geometric formulas. In this paper, we only work with finitary languages, and only nullary and binary disjunction (\(\bot \) and \(\vee \), respectively) are included in the language, which are sufficient for making up all the finitary disjunctions. Also, derivability in geometric logic is often considered in a context of a finite sequence of variables, which for example allows for an empty domain of discourse when considering semantics. We will use the term “geometric" only for reference to the finitary part of the language and logic, and consider derivability without variable contexts.

  2. This notation “\({{\,\textrm{Th}\,}}\)" is different from that used in [12], which is for sequents and not just sentences.

  3. Every \(\exists _1^+\) formula is geometric, and every gemetric formula is equivalent to a \(\exists _1^+\) formula over geometric logic, which is the weakest logic we are considering. We could simply skip this definition, and work only with geometric formulas instead. Our purpose of the definition is only to use the notation that already has been used in the literature.

  4. The reader interested in the main results of the next Sections may safely skip this Subsection, with the exception that the formula \({\textbf {A}} _{\mathrm c}\) first introduced in Proposition 3.24 will appear repeatedly later on.

  5. We can take \(C(x,y,z)\equiv 2z=(x+y)\cdot S(x+y)+2y\), and then formalize the usual arguments for the desired properties of the Cantor pairing function (totality, uniqueness and bijectivity) inside \({\mathsf I}\mathsf \Sigma _1\).

  6. The program of a partial recursive function shows how it is defined in terms of zero successor and projection functions by repeatedly applying composition, primitive recursion and minimization. Note that by choosing a suitable coding, we can assume that every natural number is the code of a program.

  7. It is true that \({\mathbb {N}}\models {\textbf {PR}} (n)\) implies \(\varphi _n\in \varvec{\mathcal{P}\mathcal{R}}\), but not vice versa. However, for every primitive recursive function f there is a natural number n such that \(\varphi _n=f\) and \({\mathbb {N}}\models {\textbf {PR}} (n)\).

  8. Note the slight difference between the current definition and the usual negative translation, in which \(A^{\mathrm g}=\lnot \lnot A\) for atomic A.

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Acknowledgements

We thank S. M. Mojtahedi for his helpful comments in our discussions, E. Jeřábek for answering our questions about models of weak theories of arithmetic and W. Ruitenberg for his comments, in particular his suggestion to us looking at the cut-off subtraction over BA.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Mohammad Ardeshir & Mohsen Shahriari

  2. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Erfan Khaniki

  3. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Erfan Khaniki

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Appendix

Appendix

1.1 The system \(\textsf{LK}\)

The system \(\textsf{LK}\) (see sections 1.2.2 and 2.3.2 of [4]) is a sequent calculus formed by the following axioms and rules: (Here, \(\Delta \), \(\Delta '\) and \(\Delta ''\) are finite lists of formulas)

1.1.1 Structural axiom

figure p

1.1.2 Structural rules

figure q

1.1.3 Logical axioms

figure r

1.1.4 Logical rules

figure s

In each rule, the formulas appearing in \(\Delta \), \(\Delta '\) or \(\Delta ''\), are called context. Other formulas which appear in the upper sequents are called active formulas, and those which appear in the lower sequents are called principal formulas. The active formula of the cut rule (Cut) is called the cut formula.

1.2 Arithmetical theories based on \(\textsf{LK}\)

Arithmetical theories over \(\textsf{LK}\) are formalized in the language of arithmetic, with the additional axioms of logic with equality (see section 2.3.3 of [4]), and axioms and rules specific to arithmetic (see section 2.4.6 of [4]):

1.2.1 Equality axioms

figure t

1.2.2 Arithmetical axioms

figure u

1.2.3 Arithmetical rule

  • , where t is substitutable for x in A, and x is not free in \(\Delta ,\Delta '\)

The formula A in the induction rule (ind) is called the induction formula. If the induction formula is restricted to a class \({\mathcal {C}}\) of formulas, the rule is called the \({\mathcal {C}}\)-induction rule.

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Ardeshir, M., Khaniki, E. & Shahriari, M. The provably total functions of basic arithmetic and its extensions. Arch. Math. Logic 64, 205–257 (2025). https://doi.org/10.1007/s00153-024-00939-0

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