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

2015
*
Gentzen's Centenary
*

We will review what is known about

doi:10.1007/978-3-319-10103-3_3
fatcat:sbpcyghcgzekjcaqawnuxnukdi
*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*...##
###
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]

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. ...

##
###
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]

2018
*
arXiv
*
pre-print

Some

arXiv:1610.02191v2
fatcat:poqgfttkgjfrxo5yvpohajq7zq
*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. ...##
###
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]

2022
*
arXiv
*
pre-print

It is

arXiv:2201.05284v2
fatcat:uo4kqvchvvav7eqlk7uonu77pu
*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 . ...##
###
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

2010
*
Bulletin of Symbolic Logic
*

There are then discussions

doi:10.2178/bsl/1286284559
fatcat:llffsna3onbpteovqxs5ki2m54
*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. ...##
###
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]

2002
*
Lecture Notes in Computer Science
*

Inductive

doi:10.1007/3-540-45470-5_3
fatcat:4bjblblahbhkrhbmrrmxx2ivsu
*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*...##
###
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]

2016
*
arXiv
*
pre-print

We determine

arXiv:1605.02534v2
fatcat:wr4chbeb5nayrd2rseyngesg4m
*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 ...##
###
Proof Theory of Martin-Löf Type Theory. An overview

2004
*
Mathématiques et sciences humaines
*

We give an overview over

doi:10.4000/msh.2959
fatcat:nf2pjysxxvho5k3ug5imbroqt4
*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*. ...
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