## Proof-theoretic analysis by iterated reflection

Venue: | Arch. Math. Logic |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Beklemishev_proof-theoreticanalysis,

author = {L. D. Beklemishev},

title = {Proof-theoretic analysis by iterated reflection},

journal = {Arch. Math. Logic},

year = {},

volume = {42},

pages = {2003}

}

### OpenURL

### Abstract

Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omega-rule. We compare the information obtained by this kind of analysis with the results obtained by the more usual proof-theoretic techniques. In some cases the techniques of iterated reflection principles allows to obtain sharper results, e.g., to define proof-theoretic ordinals relevant to logical complexity Π 0 1. We provide a more general version of the fine structure formulas for iterated reflection principles (due to U. Schmerl [24]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣn, IΣ − n, IΠ − n and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1-reflection principle for T is Σ2-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl’s theorem. 1

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Citation Context ...la [24], which is generalized and provided a new proof in this paper. The idea of using iterated reflection principles for the classification of axiomatic systems goes back to the old works of Turing =-=[27]-=- and Feferman [7]. Given a base theory T, one constructs a transfinite sequence of extensions of T by iteratedly adding formalized consistency statements, roughly, according to the following clauses: ... |

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Citation Context ...s PA. However, as is well known, the equivalence of the two systems cannot be established within PA (otherwise, PA∗ wouldhaveaspeed-upoverPA bounded by a p.t.c.f. in PA, which was disproved by Parikh =-=[18]-=-). 6 Below we analyze the situation from the point of view of proof-theoretic ordinals. Notice that PA∗ is a reasonable proof system, and it has a natural Σ1 provability predicate 2PA∗. P. Lindström [... |

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Citation Context ...generalized and provided a new proof in this paper. The idea of using iterated reflection principles for the classification of axiomatic systems goes back to the old works of Turing [27] and Feferman =-=[7]-=-. Given a base theory T, one constructs a transfinite sequence of extensions of T by iteratedly adding formalized consistency statements, roughly, according to the following clauses: (T1) T0 = T; (T2)... |

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Citation Context ...signments of fundamental sequences. 13sA slight modification of this hierarchy has recently been proposed by Weiermann and studied in detail by Möllerfeld [16]. Building on some previous results, see =-=[23]-=- for an overview, he relates this hierarchy to other natural hierarchies of function classes. Since our hierarchy has to be reasonably representable in EA, in some respects we need a sharper treatment... |

59 | The incompleteness theorems - Smorynski - 1977 |

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Citation Context ...as reasonable generalizations of the usual notions in the context of the weak arithmetic EA. The idea of using cut-free provability predicates for this kind of problems comes 35sfrom Wilkie and Paris =-=[29]-=-. Below we briefly sketch this approach and consider some typical examples. A formula ϕ is cut-free provable in a theory T (denoted T ⊢cf ϕ), if there is a finite set T0 of (closed) axioms of T such t... |

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Citation Context ... 2 ω . Together with Corollary 3.3 this implies that F(IΣ1) coincides with Fω, thatis, with the class of primitive recursive functions. This well-known result was originally established by C. Parsons =-=[19]-=-, G. Takeuti and G. Mints by other methods. 5 Extending conservation results to iterated reflection principles The definition of progressions based on iteration of reflection principles allows one to ... |

33 | Interpretability logic
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Citation Context ...es the EA-provable Σ1-completeness and has the usual provability logic — this essentially follows from the equivalence of the bounded cut-rank and the cut-free provability predicates in EA. A. Visser =-=[28]-=-, building on the work H. Friedman and P. Pudlák, established a remarkable relationship between the predicates of ordinary and cut-free provability: if T is a finite theory, then7 EA ⊢∀x(2Tϕ(˙x)↔2 cf ... |

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Citation Context ...nadequate for several reasons. The first objection is that the formula Con(T) may not be canonical, that is, it really depends on the chosen provability predicate for T rather than T itself. Feferman =-=[6]-=- gave examples of Σ1-provability predicates externally numerating PA and satisfying Löb’s derivability conditions such that the corresponding consistency assertions are not PA-provably equivalent. In ... |

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Citation Context ...+2-conservative extension of IΣ − n . From Proposition 4.5 and the characterization of induction rules in terms of reflection principles (cf. [2]) one can also obtain the following theorem of Parsons =-=[20]-=-. Corollary 4.8 For n ≥ 1, IΣn is Πn+1-conservative over IΣ R n . Proposition 4.5 also implies that IΣ1 is Π2-conservative over (EA) 2 ω . Together with Corollary 3.3 this implies that F(IΣ1) coincide... |

27 |
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Citation Context ... particular, the information of computational character. Thus, the calculation of proof-theoretic ordinals, or ordinal analysis, has become one of the central aims in the study of formal systems (cf. =-=[22]-=-). Perhaps, the most traditional approach to ordinal analysis is the definition of the proof-theoretic ordinal of a theory T as the supremum of order types of primitive recursive well-ordering relatio... |

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Citation Context ...flection principles are obtained from the above schemata by imposing a restriction that ϕ belongs to one of the classes Γ of the arithmetical hierarchy (denoted RfnΓ(T) andRFNΓ(T), respectively). See =-=[25, 13, 3]-=- for some basic information about reflection principles. We shall also consider the following metareflection rule: ϕ RRΠn(T): RFNΠn(T + ϕ) . We let Πm-RRΠn(T) denote the above rule with the restrictio... |

23 |
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Citation Context ...eas. 7sIterated consistency assertions. We deal with first order theories formulated in a language containing that of arithmetic. Our basic system is Kalmar elementary arithmetic EA (or I∆0(exp), cf. =-=[10]-=-). For convenience we assume that a symbol for the exponentiation function 2x is explicitly present in the language of EA. EA + denotes the extension of EA by an axiom stating the totality of the supe... |

10 |
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Citation Context ...cal reflection schemata. In particular, it is shown that uniform Π2-reflection principle is Σ2-conservative over the local Σ1-reflection principle. This yields as an immediate corollary the result in =-=[11]-=- on the relation between parametric and parameter-free induction schemata: IΣn is Σn+2-conservative over IΣ− n .The results of that section also provide a clear proof of a particular case of Schmerl’s... |

9 |
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Citation Context ...fine natural ordinal notation systems for suitable initial segments of the constructive ordinals, that is, they simultaneously allow for Π1 1-andΠ0 2-analyses of a theory, whenever they work. Pohlers =-=[21]-=- calls this property profoundness of the ordinal analysis. 2shierarchy) and therefore is less ad hoc. Besides, it is more robust than others, in particular, it is possible to give a formulation of the... |

9 |
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Citation Context ...ples over EA provides essentially the same information as the usual Π 0 2 -ordinal analysis. For the natural ordinal notation system up to ɛ0 similar results can be deduced from the work of R. Sommer =-=[26]-=-. Our present approach is somewhat more general and also seems to be technically simpler, so we opted for an independent presentation. Let E denote the class of elementary functions. For any set of fu... |

8 | Induction rules, reflection principles, and provably recursive functions
- Beklemishev
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(Show Context)
Citation Context ...) denotes is the elementary closure of K, that is, the closure of K∪E under composition and bounded recursion. If all the functions from K are monotone and have elementary graphs, then C(K)=E(K), see =-=[2]-=-. Let an elementary well-ordering (D, ≺) be fixed. Throughout this section we assume that there is an element 0 ∈ D satisfying EA ⊢∀α(0 = α ∨ 0 ≺ α). A hierarchy of functions Fα for α ∈ D is defined r... |

7 |
A fine structure generated by reflection formulas over Primitive Recursive Arithmetic
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(Show Context)
Citation Context ..., e.g., to define proof-theoretic ordinals relevant to logical complexity Π 0 1. We provide a more general version of the fine structure formulas for iterated reflection principles (due to U. Schmerl =-=[24]-=-). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣn, IΣ − n , IΠ − n and their combinations. We also obtain new ... |

6 |
Parameter free induction and provably total computable functions
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(Show Context)
Citation Context ...the closures of EA under Σn- andΠn+1-induction rules we have the following characterization (cf. [2]): IΣ R n ≡ IΠR n+1 ≡ (EA)n+1 ω . (E3) Parameter-free induction schemata have been characterized in =-=[5]-=-: (a) IΣ − n ≡ EA + RfnΠn Σn+1 (EA). (b) IΠ − n+1 ≡ EA + RfnΠn Σn+2 (EA). (c) EA + + IΠ − 1 ≡ EA+ + RfnΣ2(EA) ≡ EA + + RfnΣ2(EA + ). Over EA the schema IΠ − 1 is equivalent to the local Σ2-reflection ... |

5 | A proof-theoretic analysis of collection - Beklemishev - 1998 |

5 |
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(Show Context)
Citation Context ... both parts (i) and (ii), q.e.d. Proposition 7.9 EA + + IΠ − 1 is Π01-regular and |EA + + IΠ − 1 | Π0 1 = ω2 . Proof. Recall that by Proposition 6.1 (for this particular case established by Goryachev =-=[9]-=-) T + Rfn(T)isΠ1-conservative over Tω. Hence, by Theorem 4, EA + + IΠ − 1 ≡Π1 (EA + )ω ≡ ((EA) 2 1)ω ≡Π1 EA ω 2. Thus, the theory is Π 0 1-regular with the ordinal ω 2 , q.e.d. Now we consider the ext... |

3 |
Fragments of Peano's arithmetic and the MDRP theorem
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(Show Context)
Citation Context ..., S∗ β is a conservative extension of Sβ, moreover, this can be shown in EA + uniformly in β. Thus, it is sufficient to prove the lemma for the theories S∗ β . First of all, by a standard result (cf. =-=[8]-=- and [2], Proposition 5.11) based on the monotonicity of the functions Fβ we obtain that S∗ α proves induction for bounded formulas in the extended language (and this is, obviously, formalizable). 15s... |

2 |
Remarks on Magari algebras of PA and I∆0
- Beklemishev
- 1996
(Show Context)
Citation Context ...tion 4.5 If T is a Πn+1-axiomatized extension of EA, then T + RFNΣn(T) is Πn-conservative over (T) n ω. This statement has been obtained (by other methods) for T = PRA in [24], and for n =1andT=EA in =-=[1]-=-. Proof. It is sufficient to notice that (T) n ω q.e.d. 20 is closed under the rule Πn-RRΠn(T),sProposition 4.6 If U is a Πn+1-axiomatized extension of EA, then U + RFNΣn(T) is a Σn+1-conservative ext... |

2 | Notes on local reflection principles
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(Show Context)
Citation Context ...sy extension results. Throughout this section we fix an elementary well-ordering (D, ≺) and an initial elementary presented theory T. The following proposition generalizes Statement 3 of Theorem 1 in =-=[3]-=-. 21sProposition 5.1 The following statements are provable in EA: (i) ∀α RfnΣ1(T)α ⊆ Rfn(T)α; (ii) ∀α Rfn(T)α ⊆ B(Σ1) RfnΣ1(T)α. Proof. We give an informal argument by reflexive induction on α. Sinceb... |

2 |
Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt. Jahresberichte der Deutschen
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(Show Context)
Citation Context ...n (for large classes of theories) is equivalent to Π0 1-conservativity. The proposals to define general notions of proof-theoretic Π0 1-ordinals, however, generally fell victim to just criticism, see =-=[12]-=-. To refresh the reader’s memory, we discuss one such proposal below. Indoctrinated by Hilbert’s program, Gentzen formulated his ordinal analysis of Peano arithmetic as a proof of consistency of PA by... |

2 |
The optimality of induction as an axiomatization of arithmetic
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(Show Context)
Citation Context ...s more informative. In particular, this allows to obtain upper and lower bounds for proof-theoretic ordinals simultaneously. The following embedding results are known (n ≥ 1). 26s(E1) Leivant and Ono =-=[14, 17]-=- show that IΣn is equivalent to RFNΠn+2(EA) over EA, thatis, IΣn≡(EA) n+2 1 . Notice that this is sharper than the related results in [24] and the original result by Kreisel and Lévy [13] stating that... |

2 |
Reflection principles in fragments of Peano arithmetic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
- Ono
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(Show Context)
Citation Context ...s more informative. In particular, this allows to obtain upper and lower bounds for proof-theoretic ordinals simultaneously. The following embedding results are known (n ≥ 1). 26s(E1) Leivant and Ono =-=[14, 17]-=- show that IΣn is equivalent to RFNΠn+2(EA) over EA, thatis, IΣn≡(EA) n+2 1 . Notice that this is sharper than the related results in [24] and the original result by Kreisel and Lévy [13] stating that... |

1 |
The modal logic of Parikh provability
- Lindström
- 1994
(Show Context)
Citation Context ...]). 6 Below we analyze the situation from the point of view of proof-theoretic ordinals. Notice that PA∗ is a reasonable proof system, and it has a natural Σ1 provability predicate 2PA∗. P. Lindström =-=[15]-=- proves the following relationship between the provability predicates in PA and PA∗ : Lemma 9.2 EA ⊢∀x(2PA ∗(x) ↔∃n2PA2 n PA (˙x)), where 2n PA iterated 2PA. means n times Notation: The right hand sid... |

1 |
Zur Rekursion längs fundierten Relationen und Hauptfolgen
- Möllerfeld
- 1996
(Show Context)
Citation Context ...) well-ordering and does not depend on the assignments of fundamental sequences. 13sA slight modification of this hierarchy has recently been proposed by Weiermann and studied in detail by Möllerfeld =-=[16]-=-. Building on some previous results, see [23] for an overview, he relates this hierarchy to other natural hierarchies of function classes. Since our hierarchy has to be reasonably representable in EA,... |