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Proof theory and ordinal analysis

W. Pohlers
1991 Archive for Mathematical Logic  
In the first part we show why 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.  ... 
doi:10.1007/bf01621474 fatcat:xu5kxcyftnal5awg7vxhaxb7ra

Ordinal Analysis of Intuitionistic Power and Exponentiation Kripke Platek Set Theory [chapter]

Jacob Cook, Michael Rathjen
2016 Advances in Proof Theory  
With the pioneering work of Gerhard Jäger in the late 1970s 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.  ... 
doi:10.1007/978-3-319-29198-7_4 fatcat:i4pwcezft5ahxk7vnqjwwfm4ny

The Suslin operator in applicative theories: Its proof-theoretic analysis via ordinal theories

Gerhard Jäger, Dieter Probst
2011 Annals of Pure and Applied Logic  
In the context of applicative 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] .  ... 
doi:10.1016/j.apal.2011.01.009 fatcat:i7p4jlvzujdt7ktihizeu7ely4

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)

Ryota AKIYOSHI
2012 Journal of the Japan Association for Philosophy of Science  
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 ) .  ... 
doi:10.4288/kisoron.39.2_93 fatcat:eyhpcwucfbccfmid72jpejtfjm

Theories and Ordinals: Ordinal Analysis [chapter]

Michael Rathjen
2007 Lecture Notes in Computer Science  
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.  ...  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.  ... 
doi:10.1007/978-3-540-73001-9_65 fatcat:jtij35ctirffdahd7hvyh6oesm

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.  ... 

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  ... 

Admissible proof theory and beyond [chapter]

Michael Rathjen
1995 Studies in Logic and the Foundations of Mathematics  
In Section 2 I roughly describe the role of 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.  ... 
doi:10.1016/s0049-237x(06)80041-8 fatcat:gayxj5nh2zbubhudzxjkci7r2e

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.  ... 

A note on the Σ 1 spectrum of a theory

Michael Möllerfeld, Michael Rathjen
2002 Archive for Mathematical Logic  
Let T be a suitable system of classical set 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  ... 
doi:10.1007/s001530200001 fatcat:p5ctuju7ujgxhhkg2v3ebzdafm

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.  ... 

A characterization of ordinal analysis [article]

James Walsh
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.  ... 
arXiv:2112.04980v4 fatcat:vsq64elcrbderiiagjq4nidfo4

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

J. J. Joosten
2012 arXiv   pre-print
This latter observation is also seen to open up ways for applications of 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 .  ... 
arXiv:1212.2395v1 fatcat:atuc7o3babgr3jq2ysq26wd3yq
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