Interpretability in Robinson's Q

Ferreira, Fernando; Ferreira, Gilda

doi:10.2178/bsl.1903010

Cited by 7 publications

(15 citation statements)

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“…Sixth, by [36], IΣ 0 is interpretable in S 1 2 , and S 1 2 is interpretable in Q. Hence S 1 2 is essentially undecidable and mutually interpretable with Q.…”

Section: 3mentioning

confidence: 94%

“…By [36,Proposition 2,p.299], there is a bounded formula Exp(x, y, z) such that IΣ 0 proves that Exp(x, 0, z) ↔ z = 1, and Exp(x, Sy, z) ↔ ∃t(Exp(x, y, t)∧ z = t • x). However, IΣ 0 cannot prove the totality of Exp(x, y, z).…”

Section: Definition 212 (Formal Consistency and Systems)mentioning

confidence: 99%

“…From [137], PA − is interpretable in Q, and hence Q is mutually interpretable with PA − . The theory Q + is interpretable in Q (see Theorem 1 in [36], p.296), and thus mutually interpretable with Q. A. Grzegorczyk asks whether Q − is essentially undecidable.Švejdar [128] provids a positive answer to Grzegorczyk's original question by showing that Q is interpretable in Q − using the Solovay's method of shortening cuts.…”

Section: 3mentioning

confidence: 99%

“…• The theory IΣ 0 + Ω n is interpretable in Q for any n ≥ 1 (see [36,Theorem 3,p.304] The theory PA − is the theory of commutative, discretely ordered semirings with a minimal element plus the subtraction axiom. The theory PA − has the following axioms, where the language L(PA − ) is L(PA) ∪ {≤}: Definition 2.15 (The system PA − ).…”

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Current Research on Gödel’s Incompleteness Theorems

Cheng¹

2021

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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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“…Sixth, by [36], IΣ 0 is interpretable in S 1 2 , and S 1 2 is interpretable in Q. Hence S 1 2 is essentially undecidable and mutually interpretable with Q.…”

Section: 3mentioning

confidence: 94%

Section: Definition 212 (Formal Consistency and Systems)mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

mentioning

confidence: 99%

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Current Research on Gödel’s Incompleteness Theorems

Cheng¹

2021

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“…Additionally, an impressive amount of nontrivial 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)…”

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confidence: 99%

Mutual Interpretability of Robinson Arithmetic and Adjunctive Set Theory With Extensionality

Damnjanovic¹

2017

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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

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