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Proof theory and ordinal analysis

1991
*
Archive for Mathematical Logic
*

In the first part we show why

doi:10.1007/bf01621474
fatcat:xu5kxcyftnal5awg7vxhaxb7ra
*ordinals**and**ordinal*notations are naturally connected with*proof*theoretical research. We introduce the program of*ordinal**analysis*. ... The second part gives examples of applications of*ordinal**analysis*. ... We introduce this technique with the example of the*theory*of noniterated inductive definition*and*then turn to the*ordinal**analysis*of subsystems of set*theory*. ...##
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Ordinal Analysis of Intuitionistic Power and Exponentiation Kripke Platek Set Theory
[chapter]

2016
*
Advances in Proof Theory
*

With the pioneering work of Gerhard Jäger in the late 1970s

doi:10.1007/978-3-319-29198-7_4
fatcat:i4pwcezft5ahxk7vnqjwwfm4ny
*and*early 1980s, the focus switched to set*theories*, furnishing*ordinal*-theoretic*proof**theory*with a uniform*and*elegant framework. ... Until the 1970s,*proof*theoretic investigations were mainly concerned with*theories*of inductive definitions, subsystems of*analysis**and*finite type systems. ... Acknowledgement The work of the second author was supported by a Leverhulme Research Fellowship*and*the Engineering*and*Physical Sciences Research Council under grant number EP/K023128/1. ...##
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The Suslin operator in applicative theories: Its proof-theoretic analysis via ordinal theories

2011
*
Annals of Pure and Applied Logic
*

In the context of applicative

doi:10.1016/j.apal.2011.01.009
fatcat:i7p4jlvzujdt7ktihizeu7ely4
*theories*, its*proof*-theoretic strength has been analyzed in Jäger*and*Strahm (2002) [18]. ... Arai MSC: 03D70 03D75 03F03 03F10 Keywords:*Proof**theory*Suslin operator in applicative*theories**Ordinal**theories*a b s t r a c t The Suslin operator E 1 is a type-2 functional testing for the well-foundedness ... The*theory*INA of numbers*and**ordinals*In this section we introduce a*theory*of natural numbers*and**ordinals*, similar to those in [13, 17] . ...##
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Proof Theory as an Analysis of Impredicativity(New Developments in Logic: Proof-Theoretic Ordinals and Set-Theoretic Ordinals)

非可述性の分析としての証明論(<特集>あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

2012
*
Journal of the Japan Association for Philosophy of Science
*

非可述性の分析としての証明論(<特集>あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

*ordinal*

*analysis*一 93 一 ...

*Proof*

*theory*was iiiit , iELted by Hilbert for the fouエldation of mathe 皿 aticE , especially fQr car − ryillg out " Hi] bert ' g . program " , After G6del ' S incompleteness theorelns ,

*proof*

*theory*( ... ain what has been gained in

*proof*

*theory*( ')rdinal analy sis ) ． ...

##
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Theories and Ordinals: Ordinal Analysis
[chapter]

2007
*
Lecture Notes in Computer Science
*

The connection between

doi:10.1007/978-3-540-73001-9_65
fatcat:jtij35ctirffdahd7hvyh6oesm
*ordinal*representation systems*and**theories*is established in*ordinal**analysis*, a central area of*proof**theory*.*Ordinal*-theoretic*proof**theory*came into existence in 1936. ... How do*ordinals*measure the strength*and*computational power of formal*theories**and*what information can be gleaned from this correlation? This will be the guiding question of this talk. ... The connection between*ordinal*representation systems*and**theories*is established in*ordinal**analysis*, a central area of*proof**theory*.*Ordinal*-theoretic*proof**theory*came into existence in 1936. ...##
###
Page 1287 of Mathematical Reviews Vol. , Issue 84d
[page]

1984
*
Mathematical Reviews
*

*Proof*-theoretic equivalences between classical

*and*constructive

*theories*for

*analysis*, by Feferman

*and*Sieg, pp. 78-152. ... The

*ordinal*of the

*theory*W-ID, (

*proof*-theoretically equivalent to I1|-CA) is shown to be 6(Q,,-e,)0. Chapter VI. ...

##
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Page 2593 of Mathematical Reviews Vol. , Issue 96e
[page]

1996
*
Mathematical Reviews
*

Admissible

*proof**theory*, a newer trend in*ordinal**analysis*, is based on admissible sets. Pioneers of admissible*proof**theory*are Jager*and*Pohlers,*and*Rathjen is an active figure in the 1990s. ... The virtue of admissible*proof**theory*is that it provides us with a general method to approach*ordinal**analysis*, in the sense that it does not depend on specific*ordinal*systems or deduction functions. ...##
###
Page 8394 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

This is the report on

*proof**theory*, mainly*ordinal**analysis*, one has waited for. ... The main result of this paper is the determination of*proof*reduc- tion steps for a finitary*theory*of*ordinals*that is strong enough to define recursively regular*ordinals*(*and*is, in fact, equivalent ...##
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Admissible proof theory and beyond
[chapter]

1995
*
Studies in Logic and the Foundations of Mathematics
*

In Section 2 I roughly describe the role of

doi:10.1016/s0049-237x(06)80041-8
fatcat:gayxj5nh2zbubhudzxjkci7r2e
*ordinal**and**ordinal**analysis*in*proof**theory*. 1 cf.Schütte [1977], Takeuti [1987] , Pohlers [1987] , Pohlers [1991]. 2 For details cf. ...*Ordinal**analysis*has proved to be an important tool in reductive*proof**theory**and*also for the determination of the provably total functions of various complexities of a variety of*theories*. ... In Section 2 I roughly describe the role of*ordinal**and**ordinal**analysis*in*proof**theory*. Section 3 will be concerned with the program of admissible*proof**theory*as well as its achievements. ...##
###
Page 4097 of Mathematical Reviews Vol. , Issue 98G
[page]

1998
*
Mathematical Reviews
*

The reader should be familiar with first-

*and*second-order arithmetic*and*with non- standard models of arithmetic. A major concern of*proof**theory*is the*ordinal**analysis*of a given*theory*7. ... Usually a profound*proof*- theoretic*analysis*entails these four results. The main tool for computing*proof*-theoretic*ordinals*is cut elimination. ...##
###
Page 8926 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

From the text: “Contemporary

*ordinal*-theoretic*proof**theory*(i.e., the part of*proof**theory*concerned with*ordinal*analyses of strong impredicative*theories*) suffers from the extreme (*and*it seems unavoidable ... {For the entire collection see MR 2003f:03007. } 2003m:03104 03F15 03F35 Kahle, Reinhard (D-TBNG-I; Tiibingen Mathematical*proof**theory*in the light of*ordinal**analysis*. ...##
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A note on the Σ 1 spectrum of a theory

2002
*
Archive for Mathematical Logic
*

Let T be a suitable system of classical set

doi:10.1007/s001530200001
fatcat:p5ctuju7ujgxhhkg2v3ebzdafm
*theory*. We will show, that the Σ 1 spectrum of T, i.e. the set of*ordinals*having good Σ 1 definition in T is an initial segment of the*ordinals*. ... Remark: The reader familiar with*ordinal**analysis*might have expected that the spectrum of a*theory*bears interesting connections to the notation system used for its*ordinal**analysis*(of course, provided ... There is some hope that characterizations of Σ 1 spectra of intuitionistic set*theories*would contribute to a better understanding of*ordinal*informative*proof**theory**and*the role of*ordinal*representation ...##
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Page 1386 of Mathematical Reviews Vol. , Issue 82d
[page]

1982
*
Mathematical Reviews
*

Its

*ordinal*is actually known: EL + EBI, is equivalent to ID‘.,,(0), the intuitionistic*theory*of the finite constructive number classes,*and*the*proof*-theoretic*ordinal*of that*theory*is pQ.,0, by work ... The new step 82d:03090 in the*proof*of this fact is the*ordinal**analysis*of a system S *, i.e. Gédel’s*theory*of computable functions of finite type extended by bar recursion of type 0. ...##
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A characterization of ordinal analysis
[article]

2022
*
arXiv
*
pre-print

*Ordinal*

*analysis*induces a partition of Σ^1_1-definable

*and*Π^1_1-sound

*theories*whereby two

*theories*are equivalent if they have the same

*proof*-theoretic

*ordinal*. ... We show that no equivalence relation ≡ is finer than the

*ordinal*

*analysis*partition if both: (1) T≡ U whenever T

*and*U prove the same Π^1_1 sentences; (2) T≡ T+U for every set U of true Σ^1_1 sentences ... Introduction

*Ordinal*

*analysis*is a research program in

*proof*

*theory*wherein the

*proof*-theoretic

*ordinals*of

*theories*are calculated. ...

##
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Pi^0_1 ordinal analysis beyond first order arithmetic
[article]

2012
*
arXiv
*
pre-print

This latter observation is also seen to open up ways for applications of

arXiv:1212.2395v1
fatcat:atuc7o3babgr3jq2ysq26wd3yq
*ordinal**analysis*relative to some strong base*theory*. ... Moreover, we see that for an*ordinal**analysis*the full Reduction Property is not needed. ... Relative Π 0 1*ordinal**analysis*We can choose to stay within GLP ω*and*strengthen our base*theory*X . ...
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