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The Use of Trustworthy Principles in a Revised Hilbert's Program [chapter]

Anton Setzer
2015 Gentzen's Centenary  
We will review what is known about the strength up to which direct validation can be provided.  ...  Trust can be established by both providing a direct validation of such principles, which is necessarily non-mathematical and philosophical in nature, and at the same time testing those principles using  ...  Acknowledgements The author wants to thank the anonymous referees for extraordinarily detailed refereeing and many very valuable comments; Fredrik Nordvall Forsberg and Håkon Gylterud for careful proof  ... 
doi:10.1007/978-3-319-10103-3_3 fatcat:sbpcyghcgzekjcaqawnuxnukdi

Page 2739 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
In particular, the fact that the principle can be shown for trivial non-well-founded orderings like N* makes it doubtful whether the new notion is a suitable concept for defining proof theoretic strength  ...  Most theories in bounded arithmetic have the same proof-theoretic strength and therefore one needs to consider new principles in or- der to distinguish between such theories.  ... 

Proof Theory [article]

Jeremy Avigad
2017 arXiv   pre-print
Proof theory began in the 1920's as a part of Hilbert's program, which aimed to secure the foundations of mathematics by modeling infinitary mathematics with formal axiomatic systems and proving those  ...  Today such a viewpoint has applications in mathematics, computer science, and the philosophy of mathematics.  ...  ATR 0 : adds a principle of arithmetical transfinite recursion, which allows one to iterate arithmetic comprehension along countable well-orderings. 5.  ... 
arXiv:1711.01994v2 fatcat:3zjao7da6vdljgkb66w2vdjl74

Page 4097 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
4097 principle gives only a sort of plausibility before one discovers the final proof.  ...  well-ordering has order-type less then a (a so-called M}-analysis).  ... 

Proof Theory of Constructive Systems: Inductive Types and Univalence [article]

Michael Rathjen
2018 arXiv   pre-print
Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky's univalence axiom.  ...  Proof theory has contributed to a deeper grasp of the relationship between different frameworks for constructive mathematics.  ...  The main tool for performing the well-ordering proof of [34] in IARI is the following principle of transfinite recursion.  ... 
arXiv:1610.02191v2 fatcat:poqgfttkgjfrxo5yvpohajq7zq

Page 1545 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews  
weakening of proof-theoretic strength.  ...  The paper investigates the proof-theoretic strength of systems of explicit mathematics (in the sense of S.  ... 

A robust proof-theoretic well-ordering [article]

James Walsh
2022 arXiv   pre-print
It is well-known that natural axiomatic theories are pre-well-ordered by logical strength, according to various characterizations of logical strength such as consistency strength and inclusion of Π^0_1  ...  In this context these notions coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and may drop the non-mathematical quantification over "natural" theories.  ...  |T | WF is the supremum of the order-types of the primitive recursive presentations ă of well-orderings such that T $ WFpăq. |T | WF is the proof-theoretic ordinal of T .  ... 
arXiv:2201.05284v2 fatcat:uo4kqvchvvav7eqlk7uonu77pu

Page 8474 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
The author gives a proof-theoretic analysis of this subsystem of analysis. For the lower bound a well-ordering proof is given (following [T.  ...  Summary: “We analyze the proof-theoretic strength of construc- tive set theory without foundation, NCZF~, with natural numbers as urelements.  ... 

Reverse Mathematics: The Playground of Logic

Richard A. Shore
2010 Bulletin of Symbolic Logic  
There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis.  ...  While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.  ...  The basic five, in ascending order of proof theoretic strength are as follows: (RCA 0 ) Recursive Comprehension: This is a system just strong enough to prove the existence of the computable sets.  ... 
doi:10.2178/bsl/1286284559 fatcat:llffsna3onbpteovqxs5ki2m54

Page 2554 of Mathematical Reviews Vol. , Issue 95e [page]

1995 Mathematical Reviews  
More precisely the proof-theoretic strength of principles of set induction (Set-INDyn) and formula induction (Fmla-INDn) on the natural numbers, as well as an unbounded minimum oper- ator (4), is investigated  ...  Fraissé’s conjecture states that the class of linear orderings is well- quasiordered under embeddability. All its proofs known to the author use II}-CAo.  ... 

Deduction versus Computation: The Case of Induction [chapter]

Eric Deplagne, Claude Kirchner
2002 Lecture Notes in Computer Science  
Inductive proofs can be built either explicitly by making use of an induction principle or implicitly by using the so-called induction by rewriting and inductionless induction methods.  ...  We show how this applies to a uniform understanding of the so called induction by rewriting method and how this relates directly to the general use of an induction principle.  ...  Using this new framework, we uniformly review the induction by rewriting method and show how it directly relates to the induction principle, thus providing proof theoretic instead of model theoretic proofs  ... 
doi:10.1007/3-540-45470-5_3 fatcat:4bjblblahbhkrhbmrrmxx2ivsu

Page 8188 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
Feferman showed the proof-theoretic equivalence of ID, and ML, by means of an ordinal analysis (in combination with a well-ordering proof in ML,, due to Jervell and an interpre- tation of ML, in ID,,).  ...  Since P,, is easily interpretable in ML,,, the theories considered are all equivalent in terms of proof-theoretic strength (with proof-theoretic ordinal y,, where {y,: n €@} is a sequence of ordinals cofinal  ... 

Page 8395 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
(NZ-VCTR-SMC; Wellington) ; Lempp, Steffen (1-WI; Madison, WI) The proof-theoretic strength of the Dushnik-Miller theorem for countable linear orders.  ...  function parameter-free schematic versions S~ of S, thereby exhibiting different levels of strength between these principles as well as a sharp borderline be- tween fragments of analysis which are still  ... 

The strength of countable saturation [article]

B. van den Berg and E.M. Briseid and P. Safarik
2016 arXiv   pre-print
We determine the proof-theoretic strength of the principle of countable saturation in the context of the systems for nonstandard arithmetic introduced in our earlier work.  ...  What we did say is that the principle can be proved in the intuitionistic system introduced in [4] , while it adds greatly to the proof-theoretic strength of the classical system.  ...  The purpose of this short paper is to prove the first claim and to show that the addition of countable saturation to our classical system gives it the proof-theoretic strength of full second-order arithmetic  ... 
arXiv:1605.02534v2 fatcat:wr4chbeb5nayrd2rseyngesg4m

Proof Theory of Martin-Löf Type Theory. An overview

Anton Setzer
2004 Mathématiques et sciences humaines  
We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory.  ...  Then we look at the analysis of Martin-Löf type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis  ...  Note that in this proof the use of the principle of quantifier-free transfinite induction up to 0 is concentrated in the last step of the proof. Proof-theoretic strength.  ... 
doi:10.4000/msh.2959 fatcat:nf2pjysxxvho5k3ug5imbroqt4
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