## Iterated Local Reflection vs Iterated Consistency (1995)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Beklemishev95iteratedlocal,

author = {Lev Beklemishev},

title = {Iterated Local Reflection vs Iterated Consistency},

year = {1995}

}

### OpenURL

### Abstract

For "natural enough" systems of ordinal notation we show that times iterated local reflection schema over a sufficiently strong arithmetic T 0 proves the same 1-sentences as! times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at-numbers. We also derive the following more general "mixed" formulas estimating the consistency strength of iterated local reflection: for all ordinals 1 and all,

### Citations

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(Show Context)
Citation Context ...this paper we develop our notion of "natural enough" system of ordinal notation and show that such systems do exist for every recursive ordinal. 1 Introduction Since the fundamental works of=-= A.Turing [17]-=- and S.Feferman [6] transfinite recursive hierarchies of axiomatic theories have been playing a significant role in proof-theoretic studies, mainly as a kind of tool for measuring relative strength of... |

96 |
Linear Orderings
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(Show Context)
Citation Context ... ff, that is, an interpretation in PRA of the group of axioms A1 only. This can be done either by referring to the quoted theorem from [15], or, alternatively, one can use a standard theorem (cf e.g. =-=[11]-=-) stating that every recursive well-ordering can be embedded as a p.r. subset into the set of rationals Q. Then it only remains to notice that the usual ordering relation on Q is primitive recursive a... |

59 |
Subrecursion: Functions and Hierarchies
- Rose
- 1984
(Show Context)
Citation Context ...unique notation to every ordinal ! . Formally, one can define the set NF of normal forms and the ordering relation ! by simultaneous primitive recursion, e.g., in analogy with the definition given in =-=[10]-=- on page 86. (However, additional clauses corresponding to ffl-numbers will be present.) Provable linearity of the ordering (E; !E ) then guarantees that the ordering ! thus defined on NF will be prov... |

56 |
The incompleteness theorems
- Smorynski
- 1977
(Show Context)
Citation Context ...e quoted result is a corollary of a more general theorem relating different restricted versions of iterated uniform reflection principles over PRA modulo j \Pi 0 1 and j \Pi 0 n for n ? 1. Recall (cf =-=[8, 14]-=-) that the (full) uniform reflection principle for a theory T is the schema RFN(T )\Omega f8x (Pr T (pA(sx)q) ! A(x)) j A(x) 2 Fm T g ; whereas the local reflection principle is defined as Rfn(T )\Ome... |

54 |
Transfinite recursive progressions of axiomatic theories
- Feferman
- 1962
(Show Context)
Citation Context ...p our notion of "natural enough" system of ordinal notation and show that such systems do exist for every recursive ordinal. 1 Introduction Since the fundamental works of A.Turing [17] and S=-=.Feferman [6]-=- transfinite recursive hierarchies of axiomatic theories have been playing a significant role in proof-theoretic studies, mainly as a kind of tool for measuring relative strength of The research descr... |

26 |
Fragments of arithmetic
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(Show Context)
Citation Context ...alent in PRA to a \Pi 0 2 -sentence, and therefore the whole premise of the reflexive induction rule we have just proved using \Sigma 0 1 - collection is. Now, by a well-known result of C.Parsons (cf =-=[13]-=-) \Sigma 0 1 -collection schema is conservative over PRA for \Pi 0 2 -sentences, so we infer that the premise of the reflexive induction rule is provable in PRA, and we can apply this rule to get the ... |

23 |
Reflection principles and their use for establishing the complexity of axiomatic systems. Zeitschrift f. math. Logik und Grundlagen d
- Kreisel, L'evy
- 1968
(Show Context)
Citation Context ...e quoted result is a corollary of a more general theorem relating different restricted versions of iterated uniform reflection principles over PRA modulo j \Pi 0 1 and j \Pi 0 n for n ? 1. Recall (cf =-=[8, 14]-=-) that the (full) uniform reflection principle for a theory T is the schema RFN(T )\Omega f8x (Pr T (pA(sx)q) ! A(x)) j A(x) 2 Fm T g ; whereas the local reflection principle is defined as Rfn(T )\Ome... |

17 |
Solution of a problem of Leon Henkin
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(Show Context)
Citation Context ...e denoted T ff , and we shall keep the notation T ff for the hierarchy of iterated consistency assertions. We mention a few basic results on local reflection principles relevant for our work. M.H.Lob =-=[9]-=- showed that an instance Pr T (pAq) ! A of the local reflection schema is provable in the theory T if and only if so is the sentence A (Lob's Theorem). S.Feferman [6] considered transfinite recursive ... |

11 |
The formalization of interpretability
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(Show Context)
Citation Context ...e two relations V ` \Pi 0 1 U and U B V become equivalent. It is also well known that for reflexive theories the relations ` \Pi 0 1 and B are equivalent to that of relative interpretability (cf e.g. =-=[18]-=-.) Recursive Progressions. There are at least two ways to formalize the definition (T1)--(T3) of transfinite progressions of theories based on iteration of consistency. One relies on the concept of a ... |

6 |
On interpretability of some extensions of arithmetic
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- 1986
(Show Context)
Citation Context ...eflection principles. The following two auxiliary lemmas are crucial for our proof. Lemma 4.1 For any theory T , PRA proves T ! B T 1 . Proof. This fact is just a formalization of Goryachev's theorem =-=[7]-=-. For reader's convenience we give an easy informal proof and then explain why it works within PRA. The underlying idea comes from provability logic in the form of the following lemma essentially due ... |

6 |
A fine structure generated by reflection formulas over primitive recursive arithmetic
- Schmerl
- 1979
(Show Context)
Citation Context ... 0 1 -consequences of U using only specific natural ordinal notations, and a careful choice of such notations should yield proper ordinal bounds. This idea has been developed in the work of U.Schmerl =-=[12]-=- who showed among other things, e.g., that for natural ordinal notations PA j \Pi 0 1 PRA ffl 0 : Thus, ord PRA (PA) = ffl 0 , which coincides with the ordinal associated to PA through other proof-the... |

6 | Re ection principles and their use for establishing the complexity of axiomatic systems. Zeitschrift f. math. Logik und Grundlagen d - Kreisel, Levy - 1968 |

5 |
Bimodal logics for extensions of arithmetical theories. Prepublication Series LCS 7
- Beklemishev
- 1992
(Show Context)
Citation Context ...ing that the theories T 1 and T ! are, in fact, mutually interpretable, and thus, prove the same \Pi 0 1 -sentences (whatever the initial theory T , both T 1 and T ! are always reflexive theories, cf =-=[3]-=-.) In this paper we extend Goryachev's theorem to higher ordinals in the spirit of Schmerl's results, thus establishing precise relationship between the hierarchies of iterated local reflection princi... |

4 | Trans nite recursive progressions of axiomatic theories - Feferman - 1962 |

3 | On bimodal logics of provability - Beklemishev - 1994 |

3 |
Ordinal arithmetic in I \Delta 0
- Sommer
- 1993
(Show Context)
Citation Context ...n be characterized as "being essentially well-known, but not necessarily well-documented". The closest documented results of similar character that we know about are contained in the work of=-= R.Sommer [15, 16]-=-, who proves, among many other things, that the part of NWO not involving ordinal multiplication and exponentiation functions can be interpreted in PRA in this way. Moreover, he deals with much more r... |

2 |
Applications of modal logic in proof theory
- Artemov
- 1982
(Show Context)
Citation Context ... hand, Kreisel and L'evy [8] proved that T 1 = T +Rfn(T ) cannot be majorized by any recursively enumerable set of true \Pi 0 1 -sentences over T , and therefore T 1 6` T c for all c 2 O. S.N.Artemov =-=[2]-=- showed that, although obviously T ! ` T 1 , for no c 2 O with jcj ? ! do we have T c ` T 1 . This result relies on a beautiful lemma, coming from provability logic and proved by G.Boolos [5] and inde... |

2 |
Reflection principles and iterated consistency assertions
- Boolos
- 1979
(Show Context)
Citation Context ....Artemov [2] showed that, although obviously T ! ` T 1 , for no c 2 O with jcj ? ! do we have T c ` T 1 . This result relies on a beautiful lemma, coming from provability logic and proved by G.Boolos =-=[5]-=- and independently by S.N.Artemov [1], stating that in order to derive n ! ! times iterated consistency for any theory T no less than n instances of the local reflection schema for T are needed. S.Gor... |

1 |
Extensions of arithmetic and modal logics
- Artemov
- 1979
(Show Context)
Citation Context ...viously T ! ` T 1 , for no c 2 O with jcj ? ! do we have T c ` T 1 . This result relies on a beautiful lemma, coming from provability logic and proved by G.Boolos [5] and independently by S.N.Artemov =-=[1]-=-, stating that in order to derive n ! ! times iterated consistency for any theory T no less than n instances of the local reflection schema for T are needed. S.Goryachev [7] brought essentially the sa... |

1 |
Ordinal functions in fragments of arithmetic. Typeset manuscript
- Sommer
- 1992
(Show Context)
Citation Context ...n be characterized as "being essentially well-known, but not necessarily well-documented". The closest documented results of similar character that we know about are contained in the work of=-= R.Sommer [15, 16]-=-, who proves, among many other things, that the part of NWO not involving ordinal multiplication and exponentiation functions can be interpreted in PRA in this way. Moreover, he deals with much more r... |

1 | Re ection principles and iterated consistency assertions - Boolos - 1979 |

1 | 26 U.R. Schmerl. A ne structure generated by re ection formulas over Primitive Recursive Arithmetic - Orderings - 1982 |

1 | Ordinal arithmetic in I 0 - Sommer - 1992 |