## WHAT IS THE RIGHT NOTION OF SEQUENTIALITY?

### BibTeX

@MISC{Visser_whatis,

author = {Albert Visser},

title = {WHAT IS THE RIGHT NOTION OF SEQUENTIALITY?},

year = {}

}

### OpenURL

### Abstract

Abstract. In this paper we give an informally semi-rigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion m-sequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.

### Citations

131 |
Model Theory, Encyclopedia of Mathematics and its
- Hodges
- 1993
(Show Context)
Citation Context ...F M is an isomorphism between ˜ K(M) and ˜ M(M). Two interpretations K, K ′ : U → V , are i-isomorphic if there is an i-isomorphism between K and K ′ . Wilfrid Hodges calls this notion: homotopy. See =-=[Hod93]-=-, p222. The notion of i-isomorphism can be characterized in a third way. We need only demand that, for every V -model M, there is a V -formula F , which defines in M an isomorphism between ˜ K(M) and ... |

123 |
Bounded Arithmetic, Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ...ithful interpretability among finitely axiomatized sequential theories coincide. (See also [Vis93] and [Vis05].) Examples of sequential theories are Adjunctive Set Theory AS, Buss’ theory S 1 2 (see: =-=[Bus86]-=-) and synonymous variants of it like a theory of strings due to Ferreira (see: [Fer88], [Fer90], [FO06]) and a theory of sets and numbers due to Zambella (see: [Zam96]), Wilkie and Paris’ theory I∆0 +... |

72 |
Undecidable Theories
- Tarski, Mostowski, et al.
- 1953
(Show Context)
Citation Context ... Arithmetic Q in AS. Here are some historical notes about that characterization. (1) In [ST52], Wanda Szmielew and Alfred Tarski announce the interpretability of Q in AS plus extensionality. See also =-=[TMR53]-=-, p34. (2) A proof of the Szmielew-Tarski result is given by George Collins and Joseph Halpern in [CH70]. (3) Franco Montagna and Antonella Mancini, in [MM94], give an improvement of the Szmielew-Tars... |

35 | Predicative Arithmetic - Nelson |

33 | Interpretability logic
- Visser
- 1990
(Show Context)
Citation Context ...etable in adequate theories. This demand is too strong. Friedman uses the notion to provide the Friedman characterization of interpretability among finitely axiomatized sequential theories. (See also =-=[Vis90]-=- and [Vis92].) Moreover, he shows that ordinary interpretability and faithful interpretability among finitely axiomatized sequential theories coincide. (See also [Vis93] and [Vis05].) Examples of sequ... |

32 |
Consistency Statements and Interpretations
- Pudlak, \Cuts
- 1985
(Show Context)
Citation Context ...is paper [Pud83]. Pudlák uses his notion for the study of the degrees of local multi-dimensional parametric interpretability. He proves that sequential theories are prime in this degree structure. In =-=[Pud85]-=-, sequential theories provide the right level of generality for theorems about consistency statements. Let n-provability stand for provability from axioms with code below n and with a proof only invol... |

23 |
Polynomial Time Computable Arithmetic and Conservative Extensions
- Ferreira
- 1988
(Show Context)
Citation Context ...also [Vis93] and [Vis05].) Examples of sequential theories are Adjunctive Set Theory AS, Buss’ theory S 1 2 (see: [Bus86]) and synonymous variants of it like a theory of strings due to Ferreira (see: =-=[Fer88]-=-, [Fer90], [FO06]) and a theory of sets and numbers due to Zambella (see: [Zam96]), Wilkie and Paris’ theory I∆0 + Ω1 (see: [WP87]), PRA, Elementary Arithmetic EA (aka Elementary Function Arithmetic E... |

16 |
Some prime elements in the lattice of interpretability
- PUDLÁK
- 1983
(Show Context)
Citation Context ...iii) to (i) one shows that seq has an (o : o)direct interpretation in AS that is one-dimensional for the sorts s and n, using an elaborate bootstrap. We refer the reader to the literature. See, e.g., =-=[Pud83]-=-, [MPS90], [HP91]. See also Subsection 3.2 for comments on ingredients of the proof. The equivalence of (iii) and (iv) is proved in [Vis08]. ✷ We note that AS is itself sequential. Moreover, sequentia... |

16 |
The Unprovability of Small Inconsistency
- Visser
- 1991
(Show Context)
Citation Context ...sequential theories. (See also [Vis90] and [Vis92].) Moreover, he shows that ordinary interpretability and faithful interpretability among finitely axiomatized sequential theories coincide. (See also =-=[Vis93]-=- and [Vis05].) Examples of sequential theories are Adjunctive Set Theory AS, Buss’ theory S 1 2 (see: [Bus86]) and synonymous variants of it like a theory of strings due to Ferreira (see: [Fer88], [Fe... |

15 |
Polynomial time computable arithmetic
- Ferreira
- 1990
(Show Context)
Citation Context ...93] and [Vis05].) Examples of sequential theories are Adjunctive Set Theory AS, Buss’ theory S 1 2 (see: [Bus86]) and synonymous variants of it like a theory of strings due to Ferreira (see: [Fer88], =-=[Fer90]-=-, [FO06]) and a theory of sets and numbers due to Zambella (see: [Zam96]), Wilkie and Paris’ theory I∆0 + Ω1 (see: [WP87]), PRA, Elementary Arithmetic EA (aka Elementary Function Arithmetic EFA, or I∆... |

13 |
A lattice of chapters of mathematics (interpretations between theorems
- Mycielski, Pudlák, et al.
- 1990
(Show Context)
Citation Context ...i) one shows that seq has an (o : o)direct interpretation in AS that is one-dimensional for the sorts s and n, using an elaborate bootstrap. We refer the reader to the literature. See, e.g., [Pud83], =-=[MPS90]-=-, [HP91]. See also Subsection 3.2 for comments on ingredients of the proof. The equivalence of (iii) and (iv) is proved in [Vis08]. ✷ We note that AS is itself sequential. Moreover, sequentiality is p... |

11 | The formalization of interpretability - Visser - 1991 |

9 | Towards the Interpretability Logic of all Reasonable Arithmetical Theories
- Joosten
- 1998
(Show Context)
Citation Context ...sionality in Q, is given in [Nel86]. This is an interpretation with absolute identity. For further work concerning sequential theories, see, e.g., [Pud85], [Smo85], [MPS90], [HP91], [Vis93], [Vis98], =-=[JV00]-=-, [Vis05], [Vis09c], [Vis09d], [Vis09a]. 3.3. Why the Notion of Sequentiality is not Quite Right. The most direct argument that the notion of sequentiality is not quite right is to notice that there i... |

9 |
Nonstandard models and related developments
- Smorynski
- 1985
(Show Context)
Citation Context ...t U � con N n (U). This is a version of the second incompleteness theorem. The notion of sequential theory was independently invented by Friedman who called it adequate theory. See Smoryński’s survey =-=[Smo85]-=-. An important difference is that in the definition, as given by Smoryński, Elementary Arithmetic EA (aka EFA or I∆0 + exp) is stipulated to be interpretable in adequate theories. This demand is too s... |

8 | On the provability logic of bounded arithmetic - Berarducci, Verbrugge - 1993 |

8 |
Mutual Interpretability of some essentially undecidable theories
- Szmielew, Tarski
- 1950
(Show Context)
Citation Context ...n 3.1. An important part of the story of the characterization involving AS is the interpretability of Robinson’s Arithmetic Q in AS. Here are some historical notes about that characterization. (1) In =-=[ST52]-=-, Wanda Szmielew and Alfred Tarski announce the interpretability of Q in AS plus extensionality. See also [TMR53], p34. (2) A proof of the Szmielew-Tarski result is given by George Collins and Joseph ... |

6 |
A minimal predicative set theory. The Notre Dame
- Montagna, Mancini
- 1994
(Show Context)
Citation Context ...of Q in AS plus extensionality. See also [TMR53], p34. (2) A proof of the Szmielew-Tarski result is given by George Collins and Joseph Halpern in [CH70]. (3) Franco Montagna and Antonella Mancini, in =-=[MM94]-=-, give an improvement of the Szmielew-Tarski result. They prove that Q can be interpreted in an extension of AS in which we stipulate the functionality of empty set and adjoining of singletons. (4) In... |

6 |
Axiomatizability by a schema. The
- Vaught
- 1967
(Show Context)
Citation Context ...sfaction predicates and, as a consequence, restricted consistency statements. Even this answer is not quite satisfactory: in theories of pairing we also have partial satisfaction predicates. See e.g. =-=[Vau67]-=- and [Vis10]. However, these satisfaction predicates are, in a sense, more finitistic than the ones provided by sequentiality. So, we could add to our specification: these satisfaction predicates conc... |

5 |
On the interpretability of Arithmetic in Set Theory. The Notre Dame
- Collins, Halpern
- 1970
(Show Context)
Citation Context ...ielew and Alfred Tarski announce the interpretability of Q in AS plus extensionality. See also [TMR53], p34. (2) A proof of the Szmielew-Tarski result is given by George Collins and Joseph Halpern in =-=[CH70]-=-. (3) Franco Montagna and Antonella Mancini, in [MM94], give an improvement of the Szmielew-Tarski result. They prove that Q can be interpreted in an extension of AS in which we stipulate the function... |

5 |
A note on proofs of falsehood. Archiv für Mathematische Logik und Grundlagenforschung
- Krajíček
- 1987
(Show Context)
Citation Context ...he Orey-Hájek characterization and the Friedman characterization. IV. For any finitely axiomatized sequential theory U, there is an interpretation of S 1 2 in U that is Σ 0 1-sound. See e.g. [Smo85], =-=[Kra87]-=-, [Vis93], [Vis05]. V. There is a sequential theory U with p-time decidable axiom set such that the predicate logic of U is complete Π 0 2. The theory was first given in [Kra87]. The result is proved ... |

5 |
An Inside View of Exp
- Visser
- 1992
(Show Context)
Citation Context ...equate theories. This demand is too strong. Friedman uses the notion to provide the Friedman characterization of interpretability among finitely axiomatized sequential theories. (See also [Vis90] and =-=[Vis92]-=-.) Moreover, he shows that ordinary interpretability and faithful interpretability among finitely axiomatized sequential theories coincide. (See also [Vis93] and [Vis05].) Examples of sequential theor... |

4 | Modal matters in interpretability logics - Goris, Joosten - 2004 |

2 |
Oitavem I., An Interpretation of S12
- Ferreira
- 2006
(Show Context)
Citation Context ...Vis05].) Examples of sequential theories are Adjunctive Set Theory AS, Buss’ theory S 1 2 (see: [Bus86]) and synonymous variants of it like a theory of strings due to Ferreira (see: [Fer88], [Fer90], =-=[FO06]-=-) and a theory of sets and numbers due to Zambella (see: [Zam96]), Wilkie and Paris’ theory I∆0 + Ω1 (see: [WP87]), PRA, Elementary Arithmetic EA (aka Elementary Function Arithmetic EFA, or I∆0 + exp)... |

2 | The Σ0 1-conservativity of Σ0 1-completeness - Visser - 1991 |

1 |
Falsity: a study of faithful interpretations and false Σ0 1-sentences. Annals of Pure and Applied
- Faith
- 2008
(Show Context)
Citation Context ...ory AS, Buss’ theory S 1 2 (see: [Bus86]) and synonymous variants of it like a theory of strings due to Ferreira (see: [Fer88], [Fer90], [FO06]) and a theory of sets and numbers due to Zambella (see: =-=[Zam96]-=-), Wilkie and Paris’ theory I∆0 + Ω1 (see: [WP87]), PRA, Elementary Arithmetic EA (aka Elementary Function Arithmetic EFA, or I∆0 + exp), IΣ 0 1, Peano Arithmetic PA, ACA0, ZF, GB. Here are some memor... |