## Saturated models of universal theories

Venue: | Annals of Pure and Applied Logic |

Citations: | 7 - 3 self |

### BibTeX

@ARTICLE{Avigad_saturatedmodels,

author = {Jeremy Avigad},

title = {Saturated models of universal theories},

journal = {Annals of Pure and Applied Logic},

year = {},

volume = {118},

pages = {2002}

}

### OpenURL

### Abstract

A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a proof-theoretic method, Herbrand analysis, yielding uniform model-theoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the model-theoretic version. 1

### Citations

213 | Bounded Arithmetic, Propositional Logic, and Complexity Theory, volume 60 of Encyclopedia of Mathematics and Its Applications - Krajíček - 1995 |

109 |
Metamathematics of first-order arithmetic
- Hájek, Pudlák
- 1993
(Show Context)
Citation Context ... using compactness and an ultrapower construction, respectively. For these three cases, the first proof-theoretic proofs are due to Sieg. For model-theoretic proofs of the results just described, see =-=[31, 23, 24]-=-. In these examples, the relationship between the model-theoretic and prooftheoretic methods is not transparent. And while the model-theoretic methods used to obtain these results are varied (includin... |

87 | Functional interpretations of feasibly constructive arithmetic
- Cook, Urquhart
- 1993
(Show Context)
Citation Context ... � b),�a), and hence M |= ∀x ∃y ϕ(x, y,�a), as desired. � The argument for the conservation of S 1 2 over PV is similar. It is convenient to take the first-order version of PV to be the theory CPV of =-=[13]-=-, and then one only needs to show that Σ b 1 polynomial induction holds in any Herbrandsaturated model. The proof parallels the one above, except one uses bounded recursion on notations in place of pr... |

57 | Notes on polynomially bounded arithmetic
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- 1996
(Show Context)
Citation Context ...nalysis, allowing one to carry out essentially the same arguments while avoiding the use of the cut-elimination theorem. In the case of bounded arithmetic, this construction has been used in Zambella =-=[34]-=-, where it is attributed to unpublished work by Visser; see also [26, Section 7.6]. Section 4 simply notes that the general construction is widely applicable, a fact which provides uniform model-theor... |

56 |
Mathematical Intuitionism: Introduction to Proof Theory, volume 67
- Dragalin
- 1988
(Show Context)
Citation Context ...re reobtained by means of a simple forcing relation, providing alternative proofs that lie between the modeltheoretic and proof-theoretic ones. Such methods can be found in the work of Dragalin (e.g. =-=[17, 18]-=-), where they are used to obtain similar proof-theoretic 2sresults; the approach I take below stems more directly from ideas found in [2, 4, 5, 14, 15, 16]. Though the forcing constructions maintain m... |

51 | The lengths of proofs - Pudlak - 1998 |

46 |
Models of Peano Arithmetic
- Kaye
- 1991
(Show Context)
Citation Context ... using compactness and an ultrapower construction, respectively. For these three cases, the first proof-theoretic proofs are due to Sieg. For model-theoretic proofs of the results just described, see =-=[31, 23, 24]-=-. In these examples, the relationship between the model-theoretic and prooftheoretic methods is not transparent. And while the model-theoretic methods used to obtain these results are varied (includin... |

31 | Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals
- Kohlenbach
- 1996
(Show Context)
Citation Context ...2]. But the proof given in Section 3.1 of [30] is incorrect, and, indeed, there does not seem to be a way 8sof obtaining Harrington’s result using cut elimination. See Section 3 of [25] or page 69 of =-=[27]-=- for a discussion of the subtleties involved.) To prove that BΣk+1 is conservative over IΣk for Πk+2 sentences, embed IΣk in a universal theory with Skolem functions returning least witnesses to Σk fo... |

30 | First-order proof theory of arithmetic
- Buss
- 1998
(Show Context)
Citation Context ... directly to universally axiomatized theories; but by introducing appropriate Skolem functions, the methods can be used to obtain all the conservation results described above. Buss’ witnessing method =-=[9, 11]-=- is equally general, and, at the core, is very similar to Herbrand analysis. In Section 3, I will define a notion called Herbrand saturation, and I will show that every universal theory has an Herbran... |

30 |
Subsystems of Second-Order Arithmetic
- Simpson
- 1999
(Show Context)
Citation Context ... using compactness and an ultrapower construction, respectively. For these three cases, the first proof-theoretic proofs are due to Sieg. For model-theoretic proofs of the results just described, see =-=[31, 23, 24]-=-. In these examples, the relationship between the model-theoretic and prooftheoretic methods is not transparent. And while the model-theoretic methods used to obtain these results are varied (includin... |

26 |
Fragments of arithmetic
- Sieg
- 1985
(Show Context)
Citation Context ...emiregular cuts, recursive saturation, ultrapowers, and so on), it turns out that, in contrast, a single proof-theoretic method suffices throughout. Herbrand analysis, developed most fully by Sieg in =-=[29, 30]-=-, applies most directly to universally axiomatized theories; but by introducing appropriate Skolem functions, the methods can be used to obtain all the conservation results described above. Buss’ witn... |

21 |
Interpreting classical theories in constructive ones
- Avigad
(Show Context)
Citation Context .... Such methods can be found in the work of Dragalin (e.g. [17, 18]), where they are used to obtain similar proof-theoretic 2sresults; the approach I take below stems more directly from ideas found in =-=[2, 4, 5, 14, 15, 16]-=-. Though the forcing constructions maintain most of the semantic flavor of the model-theoretic ones, in Section 6, I show that instances of the algebraic proofs can be carried out in a weak constructi... |

16 | Formalizing forcing arguments in subsystems of secondorder arithmetic
- Avigad
- 1996
(Show Context)
Citation Context ...ive extension of the theory known as RCA0 , and hence of IΣ1 as well; see [31]. Syntactic proofs of Harrington’s result, involving explicit and feasible translations between theories, can be found in =-=[1]-=- and [22]. But the proof given in Section 3.1 of [30] is incorrect, and, indeed, there does not seem to be a way 8sof obtaining Harrington’s result using cut elimination. See Section 3 of [25] or page... |

16 | A new method of establishing conservativity of classical systems over their intuitionistic version
- Coquand, Hofmann
- 1999
(Show Context)
Citation Context .... Such methods can be found in the work of Dragalin (e.g. [17, 18]), where they are used to obtain similar proof-theoretic 2sresults; the approach I take below stems more directly from ideas found in =-=[2, 4, 5, 14, 15, 16]-=-. Though the forcing constructions maintain most of the semantic flavor of the model-theoretic ones, in Section 6, I show that instances of the algebraic proofs can be carried out in a weak constructi... |

12 |
Proof-theoretic equivalences between classical and constructive theories for analysis, Iterated Inductive Definitions and Subsystems of Analysis
- Feferman, Sieg
- 1981
(Show Context)
Citation Context ...sfinite induction below ω α yields a model of Πn+1 transfinite induction below α. Similar methods can be used to obtain the conservation results of Friedman [20], along the lines of Feferman and Sieg =-=[19]-=-. For another model-theoretic approach to ordinal analysis, see [6]. 9s5 An algebraic version There is a more direct way of obtaining the model M constructed in the proof of Theorem 3.2: given Sω, let... |

11 | A model-theoretic approach to ordinal analysis
- Avigad, Sommer
- 1997
(Show Context)
Citation Context ...tion below α. Similar methods can be used to obtain the conservation results of Friedman [20], along the lines of Feferman and Sieg [19]. For another model-theoretic approach to ordinal analysis, see =-=[6]-=-. 9s5 An algebraic version There is a more direct way of obtaining the model M constructed in the proof of Theorem 3.2: given Sω, let ˆ S be a maximally consistent extension, and “read off” a model fr... |

10 |
Remarks on Herbrand normal forms and Herbrand realizations
- Kohlenbach
- 1992
(Show Context)
Citation Context ...found in [1] and [22]. But the proof given in Section 3.1 of [30] is incorrect, and, indeed, there does not seem to be a way 8sof obtaining Harrington’s result using cut elimination. See Section 3 of =-=[25]-=- or page 69 of [27] for a discussion of the subtleties involved.) To prove that BΣk+1 is conservative over IΣk for Πk+2 sentences, embed IΣk in a universal theory with Skolem functions returning least... |

8 | Algebraic proofs of cut elimination
- Avigad
(Show Context)
Citation Context .... Such methods can be found in the work of Dragalin (e.g. [17, 18]), where they are used to obtain similar proof-theoretic 2sresults; the approach I take below stems more directly from ideas found in =-=[2, 4, 5, 14, 15, 16]-=-. Though the forcing constructions maintain most of the semantic flavor of the model-theoretic ones, in Section 6, I show that instances of the algebraic proofs can be carried out in a weak constructi... |

8 |
Interpretability and fragments of arithmetic
- Hájek
- 1993
(Show Context)
Citation Context ...nsion of the theory known as RCA0 , and hence of IΣ1 as well; see [31]. Syntactic proofs of Harrington’s result, involving explicit and feasible translations between theories, can be found in [1] and =-=[22]-=-. But the proof given in Section 3.1 of [30] is incorrect, and, indeed, there does not seem to be a way 8sof obtaining Harrington’s result using cut elimination. See Section 3 of [25] or page 69 of [2... |

8 |
Herbrand analyses
- Sieg
- 1991
(Show Context)
Citation Context ...emiregular cuts, recursive saturation, ultrapowers, and so on), it turns out that, in contrast, a single proof-theoretic method suffices throughout. Herbrand analysis, developed most fully by Sieg in =-=[29, 30]-=-, applies most directly to universally axiomatized theories; but by introducing appropriate Skolem functions, the methods can be used to obtain all the conservation results described above. Buss’ witn... |

5 |
A proof-theoretic analysis of collection
- Beklemishev
- 1998
(Show Context)
Citation Context ...than a; see [23, Section 1.63]. (The argument in [29] is not quite right, but can be repaired along the lines just sketched. For other proof-theoretic proofs of this conservation result, see [10] and =-=[8]-=-.) Finally, to prove that Σ 1 1 -AC0 is conservative over Peano Arithmetic, embed PA in a second-order universal theory with function symbols. In this theory, allow operations on the function sorts th... |

5 | The witness function method and provably recursive functions of Peano arithmetic
- Buss
- 1991
(Show Context)
Citation Context ...f x less than a; see [23, Section 1.63]. (The argument in [29] is not quite right, but can be repaired along the lines just sketched. For other proof-theoretic proofs of this conservation result, see =-=[10]-=- and [8].) Finally, to prove that Σ 1 1 -AC0 is conservative over Peano Arithmetic, embed PA in a second-order universal theory with function symbols. In this theory, allow operations on the function ... |

4 | An application of constructive completeness
- COQUAND, SMITH
- 1996
(Show Context)
Citation Context |

3 |
Transfer principles for intuitionistic nonstandard arithmetic
- Avigad, Helzner
(Show Context)
Citation Context |

3 |
On Recursively Saturated Models of Arithmetic
- Barwise, Schlipf
(Show Context)
Citation Context ... Wilkie. The other results were first obtained using model-theoretic methods. The conservation of WKL0 over PRA is due to Friedman. The conservation of Σ 1 1 -AC over PA is due to Barwise and Schlipf =-=[7]-=-, using recursively saturated models. Finally, the conservation of BΣk+1 over IΣk is due to Friedman and Paris independently, using compactness and an ultrapower construction, respectively. For these ... |

2 | Ordinal analysis without proofs
- Avigad
- 2002
(Show Context)
Citation Context ... , tk it is not difficult to obtain a term s such that M satisfies ∀x ϕ(x, sx), as required. It seems worth mentioning that by combining the notion of Herbrand saturation with the methods of [10] and =-=[3]-=- one can carry out the ordinal analysis of, say, Peano arithmetic, without relying on cut-elimination. For example, if α is infinite and closed under multiplication, an Herbrand-saturated model of a s... |

1 | Two applications of boolean models
- Coquand
- 1997
(Show Context)
Citation Context |

1 |
Explict algebraic models for constructive and classical theories with nonstandard elements
- Dragalin
- 1995
(Show Context)
Citation Context ...re reobtained by means of a simple forcing relation, providing alternative proofs that lie between the modeltheoretic and proof-theoretic ones. Such methods can be found in the work of Dragalin (e.g. =-=[17, 18]-=-), where they are used to obtain similar proof-theoretic 2sresults; the approach I take below stems more directly from ideas found in [2, 4, 5, 14, 15, 16]. Though the forcing constructions maintain m... |

1 |
Iterated inductive definitions and Σ 1 1-AC
- Friedman
- 1970
(Show Context)
Citation Context ...of a suitable Skolemized version of Πn transfinite induction below ω α yields a model of Πn+1 transfinite induction below α. Similar methods can be used to obtain the conservation results of Friedman =-=[20]-=-, along the lines of Feferman and Sieg [19]. For another model-theoretic approach to ordinal analysis, see [6]. 9s5 An algebraic version There is a more direct way of obtaining the model M constructed... |

1 |
Finitist proofs of conservation. Posting to the Foundations of Mathematics forum
- Friedman
- 1999
(Show Context)
Citation Context ...e Σ1 soundness of ERA, there really is such a proof. � 1 Another method of obtaining finitary proofs of conservation results like the ones we have been studying has recently been sketched by Friedman =-=[21]-=-. 14sNotes. 1. In fact, the methods of Section 5 yield constructive proofs, which is to say, in the proof of Theorem 6.2 it is sufficient to use ERA with firstorder intuitionistic logic. With that res... |