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We use model-theoretic methods described in  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  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,  ). ...doi:10.2307/2586768 fatcat:iwy7zvmhtnde5mofb7wiivcs54
We use model-theoretic methods described in  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 ...doi:10.1184/r1/6492848.v1 fatcat:2ljag7yoqrh6zkcsm4xrvj3wuu
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 ...
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. ...
Moreover, we establish that also its schematic extension -- in 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. ...arXiv:2007.07188v3 fatcat:s3r5tmrnhfbklh4xr3jngcdvdu
This paper gives a detailed account 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. ...arXiv:1807.10022v1 fatcat:z6xtuwgiofdwfptqvaixohsaiu
Advances in Proof Theory
Independently 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. ...doi:10.1007/978-3-319-29198-7_7 fatcat:qfyqdz55izanzpxktug24lqjnm
To this end, 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]. ...doi:10.1016/j.apal.2013.05.010 fatcat:ryogucyqyvhofpfrqc5z37nru4
We argue that any natural axiomatization 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 . ...doi:10.2178/jsl/1146620166 fatcat:g76afpuxhjez3azvmfnpx3jipy
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 ...
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. ...
This paper deals with 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 . ...doi:10.1016/s0168-0072(00)00016-6 fatcat:muzmf3pywnbhpirrazimb4qc3m
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. ...
The Oxford Handbook of Philosophy of Mathematics and Logic
reducible to 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. ...doi:10.1093/0195148770.003.0019 fatcat:2vqhi4dnwbgprpb7dawngvgjjm
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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