Kristiansen and Murwanashyaka recently proved that Robinson arithmetic, Q, is interpretable in an elementary theory of full binary trees, T. We prove that, conversely, T is interpretable in Q by producing a formal interpretation of T in an elementary concatenation theory QT+, thereby also establishing mutual interpretability of T with several well-known weak essentially undecidable theories of numbers, strings, and sets. We also introduce a “hybrid” elementary theory of strings and trees, WQT*, and establish its mutual interpretability with Robinson’s weak arithmetic R, the weak theory of trees WT of Kristiansen and Murwanashyaka, and the weak concatenation theory WTCε of Higuchi and Horihata.

An elementary theory of concatenation, QT + , is introduced and used to establish mutual interpretability of Robinson arithmetic, Minimal Predicative Set Theory, quantifier-free part of Kirby's finitary set theory, and Adjunctive Set Theory, with or without extensionality.] arithmetical laws. Additionally, an impressive amount of non-trivial mathematics can be reconstructed in theories interpretable in Q, including (first-order) Euclidean geometry, elementary theory of the real closed fields (i.e., first-order theory of real numbers) as well as basic "feasible analysis" formalizing elementary properties of real numbers and continuous functions.(See [4] for details.) It is frequently pointed out that Q is a minimal element in the well-ordered hierarchy of interpretability of "natural" mathematical theories.It was Tarski who first noted that, as regards self-referential constructions at the heart of meta-mathematical arguments for incompleteness, the procedure of arithmetization by means of which the syntax of formal theories is coded up by numbers amounts to an unnecessary detour. In his seminal work on the concept of truth of formalized languages Tarski introduced a theory of concatenated strings to demonstrate this point. This idea was further developed by Quine [13]. More recently, Grzegorczyk has suggested that a theory of concatenated "texts" would form a natural framework for the study of incompleteness phenomena and, more generally, computation, and for this purpose he introduced a weak theory of concatenation, TC, and proved its

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered:1. There exist sentences that are neither paradoxical nor grounded.2. There are fixed points.3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n ∣ A(n) is true in the minimal fixed point}, where A(x) is a formula of AR + T) are precisely the sets.4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the sets.5. The closure ordinal for Kripke's construction of the minimal fixed point is .In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen:1. There may or may not exist nonparadoxical, ungrounded sentences.2. The number of fixed points may be any positive finite number, ℵ0, or .3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the sets, including intermediate cases.4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the sets.5. The closure ordinal for the construction of the minimal fixed point may be ω, , or any successor limit ordinal in between.In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Gödel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language.

A realizability notion that employs only primitive recursive functions is defined, and, relative to it, the soundness of the fragment of Heyting Arithmetic (HA) in which induction is restricted to formulae is proved. A dual concept of falsifiability is proposed and an analogous soundness result is established for a further restricted fragment of HA.