## WHY THE THEORY R IS SPECIAL (2009)

### BibTeX

@MISC{Visser09whythe,

author = {Albert Visser},

title = {WHY THE THEORY R IS SPECIAL},

year = {2009}

}

### OpenURL

### Abstract

Abstract. Is it possible to give coordinate-free characterizations of salient theories? Such characterizations would always involve some notion of sameness of theories: we want to describe a theory modulo a notion of sameness, without having to give an axiomatization in a specific language. Such a characterization could, e.g., be a first order formula in the language of partial preorderings that describes uniquely a degree in a particular structure of degrees of interpretability. Our theory would be contained in this degree. There are very few examples currently known along these lines, except some rather trivial ones. In this paper we provide a non-trivial characterization of Tarski-Mostowski-Robinson’s theory R. The characterization is in terms of the double degree structure of RE degrees of local and global interpretability. Consider the RE degrees of global interpretability that are in the minimal RE degree of local interpretability. These are the global degrees of the RE locally finitely satisfiable theories. We show that these degrees have a maximum and that R is in that maximum. In more mundane terms: an RE theory is locally finite iff it is globally interpretable in R. Dedicated to Harvey Friedman on the occasion of his 60th birthday. 1.

### Citations

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(Show Context)
Citation Context ...egree. 3 As we will see this means that an RE theory is locally finitely satisfiable iff it is globally interpretable in R. The theory R was introduced by Tarski, Mostowski and Robinson in their book =-=[TMR53]-=-. It is a very weak theory that is essentially undecidable. This means that every consistent RE extension of the theory is undecidable. It was observed by Cobham that one still has an essentially unde... |

34 |
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(Show Context)
Citation Context ...ce in R. Robinson’s Arithmetic Q was introduced in [TMR53]. Using Solovay’s method of shortening cuts (see [Solle]), one can show that Q interprets seemingly much stronger theories like I∆0 + Ω1. See =-=[Nel86]-=- and [HP91]. Here are the axioms of Q. Q1. ⊢ Sx = Sy → x = y Q2. ⊢ 0 ̸= Sx Q3. ⊢ x = 0 ∨ ∃y x = Sy Q4. ⊢ x + 0 = x Q5. ⊢ x + Sy = S(x + y) Q6. ⊢ x × 0 = 0 Q7. ⊢ x × Sy = x × y + x The theory Q − is du... |

13 |
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(Show Context)
Citation Context ...he signature nor on the complexity of the theory. The intended notion of interpretation is more-dimensional interpretation with parameters. This structure is studied by Mycielski, Pudlák and Stern in =-=[MPS90]-=-. They call these degrees: chapters. The maximum of the 2000 Mathematics Subject Classification. 03A05, 03B30, 03F25, Key words and phrases. interpretability, local interpretability, finite satisfiabi... |

6 |
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(Show Context)
Citation Context ...sistent RE extension of the theory is undecidable. It was observed by Cobham that one still has an essentially undecidable theory if one drops the axiom R6 (given below), obtaining the theory R0. See =-=[Vau62]-=- and [JS83]. Cobham has shown that R has a stronger property than essential undecidability. Consider any RE theory T . Suppose we have translation α of the arithmetical language into the language of T... |

5 | An interpretation of Robinson’s Arithmetic in Grzegorczyk’s weaker variant - unknown authors |

5 | Can we make the Second Incompleteness Theorem coordinate free
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- 2009
(Show Context)
Citation Context ...ment to build an interpretation of T in S1 2. (See e.g. [Vis08].) In terms of that paper, we have shown: Q ✄ ✵(T ).) This interpretation is one-dimensional, one-piece and parameter free. Moreover, by =-=[Vis09b]-=-, Corollary 6.1, we find that we can make the interpretation identity-preserving. 6 In fact, the argument is closely analogous with the construction of a Σ 0 1 -truth predicate that works for formulas... |

4 |
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(Show Context)
Citation Context ...ct R is interpretable in R0. In [Vis09a], it is shown that R⋆ is interpretable in R0, and hence in R. Robinson’s Arithmetic Q was introduced in [TMR53]. Using Solovay’s method of shortening cuts (see =-=[Solle]-=-), one can show that Q interprets seemingly much stronger theories like I∆0 + Ω1. See [Nel86] and [HP91]. Here are the axioms of Q. Q1. ⊢ Sx = Sy → x = y Q2. ⊢ 0 ̸= Sx Q3. ⊢ x = 0 ∨ ∃y x = Sy Q4. ⊢ x ... |

3 |
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(Show Context)
Citation Context ...xtension of the theory is undecidable. It was observed by Cobham that one still has an essentially undecidable theory if one drops the axiom R6 (given below), obtaining the theory R0. See [Vau62] and =-=[JS83]-=-. Cobham has shown that R has a stronger property than essential undecidability. Consider any RE theory T . Suppose we have translation α of the arithmetical language into the language of T . Suppose ... |