Weak essentially undecidable theories of concatenation

Abstract

In the language $$\lbrace 0, 1, \circ , \preceq \rbrace $$ { 0 , 1 , ∘ , ⪯ } , where 0 and 1 are constant symbols, $$\circ $$ ∘ is a binary function symbol and $$\preceq $$ ⪯ is a binary relation symbol, we formulate two theories, $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D , that are mutually interpretable with the theory of arithmetic $$ {\textsf {R}} $$ R and Robinson arithmetic $${\textsf {Q}} $$ Q , respectively. The intended model of $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D is the free semigroup generated by $$\lbrace {\varvec{0}}, {\varvec{1}} \rbrace $$ { 0 , 1 } under string concatenation extended with the prefix relation. The theories $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D are purely universally axiomatised, in contrast to $$ {\textsf {Q}} $$ Q which has the $$\varPi _2$$ Π 2 -axiom $$\forall x \; [ \ x = 0 \vee \exists y \; [ \ x = Sy \ ] \ ] $$ ∀ x [ x = 0 ∨ ∃ y [ x = S y ] ] .

A weak theory of building blocks

“…In Ganea [7], Švejdar [22], and Visser [26], it is shown that 𝖳𝖢 is mutually interpretable with 𝖰. In [14], we prove mutual interpretability of 𝖳𝖢 and a fragment whose intended model is the free semigroup of binary strings extended with the prefix relation. The binary operation of pairing is essential for encoding other mathematical objects like tuples and sequences of arbitrary length.…”

“…In Ganea [7], Švejdar [22], and Visser [26], it is shown that $$\mathsf {TC}$$ is mutually interpretable with $$\mathsf {Q}$$. In [14], we prove mutual interpretability of $$\mathsf {TC}$$ and a fragment whose intended model is the free semigroup of binary strings extended with the prefix relation.…”

Pour-El’s Landscape

“…One can show that Succ • is decidable and that every sentence is equivalent to a Boolean combination of sentences C n saying 'there is a cycle of size n'. See, e.g., [12,Appendix A]. We note that, over Succ • , the C n are mutually independent.…”

“…⊣ The relationships between these non-effective notions are visualised in Figure 1. In [12], the existence of a decidable f-essentially incomplete theory was proved. Also, essential hereditary undecidability and computable inseparability are incomparable in general (cf.…”

On Interpretability Between Some Weak Essentially Undecidable Theories

“…These are natural theories given by a handful of transparent axioms. They are all mutually interpretable with each other and also with a number of other natural theories, e.g., the tree theories studied in Kristiansen & Murwanashyaka [9], Damnjanovic [2] [3] and Murwanashyaka [11], and the concatenation theories studied in Murwanashyaka [10].…”