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The model-theoretic ordinal analysis of theories of predicative strength

1999
*
Journal of Symbolic Logic (JSL)
*

We use

doi:10.2307/2586768
fatcat:iwy7zvmhtnde5mofb7wiivcs54
*model*-*theoretic*methods described in [3] to obtain*ordinal*analyses*of*a number*of**theories**of*first- and second-order arithmetic, whose proof-*theoretic**ordinals*are less than or equal to Γ0. ... Introduction In [3] we introduced a*model*-*theoretic*approach to*ordinal**analysis*as an interesting alternative to cut elimination. ... Work*of*Feferman and Schütte has established that*the**ordinal*Γ 0 is*the*least upper bound to*the**strength**of*such*theories*(see for example, [4] ). ...##
###
The Model-Theoretic Ordinal Analysis of Theories of Predicative Strength

2018

We use

doi:10.1184/r1/6492848.v1
fatcat:2ljag7yoqrh6zkcsm4xrvj3wuu
*model*-*theoretic*methods described in [3] to obtain*ordinal*analyses*of*a number*of**theories**of*first- and second-order arithmetic. ... For*the**analysis**of**predicative**theories*we need a sufficiently strong notation system. ... To augment our notation, we add to*the*assertion A -X# (H)*the*information that A is ax-large by writing A cX Z(H)*The*following lemma can be thought*of*as a*model*-*theoretic*counterpart to*the**predicative*...##
###
Page 4097 of Mathematical Reviews Vol. , Issue 98G
[page]

1998
*
Mathematical Reviews
*

In a second pa- per [“

*The**model*-*theoretic**ordinal**analysis**of**theories**of*pred- icative*strength*”, J. ... This paper describes a*model*-*theoretic*approach to*the**ordinal**analysis**of**predicative*subsystems*of*first- and second-order arith- metic, namely*the**theories*1Z;, RCAy, WK Ly, IZ,, PA, ACAy and x}-ACy ...##
###
Page 6394 of Mathematical Reviews Vol. , Issue 91M
[page]

1991
*
Mathematical Reviews
*

This impressive dissertation analyses

*the**strength**of*several im-*predicative*subsystems*of*second-order number*theory*by deter- mining their proof-*theoretic**ordinals*. ... For any such*theory*T its proof-*theoretic**ordinal*|7| is defined to be*the*least*ordinal*a such that a cannot be*the**ordinal**of*a provable primitive recursive well- ordering in 7. ...##
###
Nonclassical truth with classical strength. A proof-theoretic analysis of compositional truth over HYPE
[article]

2020
*
arXiv
*
pre-print

Moreover, we establish that also its schematic extension -- in

arXiv:2007.07188v3
fatcat:s3r5tmrnhfbklh4xr3jngcdvdu
*the*sense*of*Feferman -- is as strong as*the*schematic extension*of*KF, thus matching*the**strength**of**predicative**analysis*. ... Questions concerning*the*proof-*theoretic**strength**of*classical versus non-classical*theories**of*truth have received some attention recently. ... Theorem 1 is key to our proof-*theoretic**analysis**of*a*theory**of*truth over HYPE. We now turn to*the*definition*of*such a truth*theory*. ...##
###
Computational reverse mathematics and foundational analysis
[article]

2018
*
arXiv
*
pre-print

This paper gives a detailed account

arXiv:1807.10022v1
fatcat:z6xtuwgiofdwfptqvaixohsaiu
*of**the*motivations and methodology*of*foundational*analysis*, which have heretofore been largely left implicit in*the*practice. ... Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in*the*same way as inequivalences, namely by considering only ω-*models**of**the*systems in question. ... I would also like to thank Kentaro Fujimoto, who helped me determine*the*correct base*theory*for lemma 7.10, and François G. ...##
###
The Operational Perspective: Three Routes
[chapter]

2016
*
Advances in Proof Theory
*

Independently

doi:10.1007/978-3-319-29198-7_7
fatcat:qfyqdz55izanzpxktug24lqjnm
*of*that work, in (F 1971) I had made use*of*extensions*of*Gödel's functional ("Dialectica") interpretation to determine*the*proof-*theoretical**strength**of*various subsystems*of**analysis*by ... In*the*years following*the*characterization (F 1964, Schütte 1965)*of**predicative**analysis*in terms*of*an autonomous progression*of*ramified analytic systems whose limit is at*the**ordinal*Γ 0 , I had explored ... I wish to thank Ulrik Buchholtz, Gerhard Jäger, Dieter Probst, Michael Rathjen, and Thomas Strahm for their helpful comments on a draft*of*this article. ...##
###
A proof-theoretic account of classical principles of truth

2013
*
Annals of Pure and Applied Logic
*

To this end,

doi:10.1016/j.apal.2013.05.010
fatcat:ryogucyqyvhofpfrqc5z37nru4
*the*proof-*theoretic**strength**of*a number*of*axiomatic*theories**of*truth over intuitionistic logic is determined. ...*The**theories*considered correspond to*the*maximal consistent collections*of*fifteen truththeoretic principles as isolated in Leigh and Rathjen 2012. ... Acknowledgements This work was supported by*the*Arts and Humanities Research Council UK [AH/H039791/1]. ...##
###
Axiomatizing Kripke's theory of truth

2006
*
Journal of Symbolic Logic (JSL)
*

We argue that any natural axiomatization

doi:10.2178/jsl/1146620166
fatcat:g76afpuxhjez3azvmfnpx3jipy
*of*Kripke's*theory*in Strong Kleene logic has*the*same proof-*theoretic**strength*as PKF. namely*the**strength**of**the*system ramified*analysis*or a system*of*Tarskian ... We investigate axiomatizations*of*Kripke's*theory**of*truth based on*the*Strong Kleene evaluation scheme for treating sentences lacking a truth value. ... We determine*the*proof-*theoretic**strength**of*PKF as that*of*ramified*analysis*RA <ω ω up to ω ω . ...##
###
Page 3013 of Mathematical Reviews Vol. , Issue 84h
[page]

1984
*
Mathematical Reviews
*

*The*introductory section (“Why

*ordinal*

*analysis*?) discusses

*the*proof-

*theoretic*

*ordinal*

*of*a system T and reasons for being interested in it. ... From

*the*summary: “This paper provides an account

*of*an approach to

*modeling*unknown data by means

*of*fuzzy set

*theory*and addresses

*the*problem

*of*deriving

*the*uncertainty on a sum

*of*variables whose ...

##
###
Page 3868 of Mathematical Reviews Vol. , Issue 2000f
[page]

2000
*
Mathematical Reviews
*

*ordinal*

*analysis*

*of*

*theories*

*of*

*predicative*

*strength*. ... Cut elimination has been a powerful tool for

*ordinal*

*analysis*. On

*the*other hand,

*the*authors introduced a

*model*-

*theoretic*approach to

*ordinal*

*analysis*in their 1997 paper [Bull. ...

##
###
The non-constructive μ operator, fixed point theories with ordinals, and the bar rule

2000
*
Annals of Pure and Applied Logic
*

This paper deals with

doi:10.1016/s0168-0072(00)00016-6
fatcat:muzmf3pywnbhpirrazimb4qc3m
*the*proof*theory**of*ÿrst-order applicative*theories*with non-constructive operator and a form*of**the*bar rule, yielding systems*of**ordinal**strength*0 and '20, respectively. ... Relevant use is made*of*ÿxed-point*theories*with*ordinals*plus bar rule. ... We will show that*the*corresponding*theory*AutBON( ) has*the*same proof-*theoretic**strength*as*predicative**analysis*and, hence, its proof-*theoretic**ordinal*is exactly*the*Feferman-Sch utte*ordinal*0 . ...##
###
Page 8927 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

For a formal

*theory**of*truth, its*strength*as a*theory**of*truth and its proof-*theoretic**strength*are not necessarily parallel to each other. ... Here, as an example, a*theory**of*self-referential truth which is weak as a*theory**of*truth is shown to have*the*same proof-*theoretic**strength*as its more elaborate parent system. ...##
###
Predicativity
[chapter]

2005
*
The Oxford Handbook of Philosophy of Mathematics and Logic
*

reducible to

doi:10.1093/0195148770.003.0019
fatcat:2vqhi4dnwbgprpb7dawngvgjjm
*the*union*of*all these systems. 9 If T has*the*same proof-*theoretic**strength*as that progression, then its proof-*theoretic**ordinal*is Γ 0 . ... such*ordinals*might well be used metamathematically in an evaluation*of**the*proof-*theoretical**strength**of**the*system F proposed to represent P. ...##
###
Page 5912 of Mathematical Reviews Vol. , Issue 92k
[page]

1992
*
Mathematical Reviews
*

*The*

*ordinal*

*analysis*stops at

*the*systems KP! and KPi

*of*admissible set

*theory*, which are proof-

*theoretically*weaker than

*analysis*(second-order arithmetic) with II}-comprehension. ... This survey

*of*

*ordinal*

*analysis*and proof-

*theoretic*

*ordinals*sum- marizes developments after a book by

*the*author [Proof

*theory*, Lec- ture Notes in Math., 1407, Springer, Berlin, 1989; MR 91h:03078] and ...

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