Weak theories of concatenation and minimal essentially undecidable theories

Higuchi, Kojiro; Horihata, Yoshihiro

doi:10.1007/s00153-014-0391-x

Cited by 7 publications

(8 citation statements)

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“…This was further improved in Ganea [3], Visser [19] and Švejdar [16] where it was shown that TC is mutually interpretable with Robinson arithmetic. In Higuchi and Horihata [6], it was shown that TC is minimal essentially undecidable. That is, removing any one of the axioms of TC gives a theory that is not essentially undecidable.…”

Section: The Axioms Of Wdmentioning

confidence: 99%

Weak essentially undecidable theories of concatenation

Murwanashyaka

2022

Arch. Math. Logic

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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 ] ] .

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Section: The Axioms Of Wdmentioning

confidence: 99%

Weak essentially undecidable theories of concatenation

Murwanashyaka

2022

Arch. Math. Logic

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“…Kojiro Higuchi and Yoshihiro Horihata introduce the theory of concatenation WTC − , which is a weak subtheory of Grzegorczyk's theory TC, and show that WTC − is minimal essentially undecidable and WTC − is mutually interpretable with R (see [60]).…”

Section: 3mentioning

confidence: 99%

Current Research on Gödel’s Incompleteness Theorems

Cheng¹

2021

Bull. symb. log

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We give a survey of current research on Gödel's incompleteness theorems from the following three aspects: classifications of different proofs of Gödel's incompleteness theorems, the limit of the applicability of Gödel's first incompleteness theorem, and the limit of the applicability of Gödel's second incompleteness theorem. §1. Introduction. Gödel's first and second incompleteness theorems are some of the most important and profound results in the foundations of mathematics and have had wide influence on the development of logic, philosophy, mathematics, computer science, as well as other fields. Intuitively speaking, Gödel's incompleteness theorems express that any rich enough logical system cannot prove its own consistency, i.e., that no contradiction like 0 = 1 can be derived within this system.Gödel [43] proves his first incompleteness theorem (G1) for a certain formal system P related to Russell-Whitehead's Principia Mathematica based on the simple theory of types over the natural number series and the Dedekind-Peano axioms (see [8, p. 3]). Gödel announces the second incompleteness theorem (G2) in an abstract published in October 1930: no consistency proof of systems such as Principia, Zermelo-Fraenkel set theory, or the systems investigated by Ackermann and von Neumann is possible by methods which can be formulated in these systems (see [147, p. 431]).Gödel comments in a footnote of [43] that G2 is corollary of G1 (and in fact a formalized version of G1): if T is consistent, then the consistency of T is not provable in T where the consistency of T is formulated as the arithmetic formula which says that there exists an unprovable sentence in T. Gödel [43] sketches a proof of G2 and promises to provide full details in a subsequent publication. This promise is not fulfilled, and a detailed proof of G2 for first-order arithmetic only appears in a monograph by Hilbert and Bernays [59]. Abstract logic-free formulations of Gödel's incompleteness theorems have been given by Kleene [76] ("symmetric form"), Smullyan [121] ("representation systems"), and others. The following is a modern reformulation of Gödel's incompleteness theorems.

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“…A rather recent line of research in concatenation theory has focused on interpretablity between weak first-order theories and essential undecidability. Grzegorczyk [5], Grzegorczyk and Zdanowski [6], Svejdar [22], Visser [24], Horihata [11] and Higuchi and Horihata [10] belong to this line. Another recent line of research has focused on word equations and formal languages.…”

Section: References and Related Researchmentioning

confidence: 99%