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Page 3413 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
Gylys (Vilnius) 97f:03075 03F30 03C62 Sommer, Richard (1-STF; Stanford, CA) Transfinite induction within Peano arithmetic. (English summary) Ann. Pure Appl. Logic 76 (1995), no. 3, 231-289.  ...  He keeps within Peano arithmetic and so limits himself to induction up to ordinals strictly less than €9 for formulas of class £,, or [,,.  ... 

Page 7202 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
Sommer, “Transfinite induction and hierarchies generated by transfinite recursion within Peano arithmetic”, Ph.D. Thesis, Univ. California, Berkeley, CA, 1990].  ...  96m:03036 96m:03036 03F30 03D20 Buss, Samuel R. (1-UCSD; La Jolla, CA) The witness function method and provably recursive functions of Peano arithmetic.  ... 

Page 3957 of Mathematical Reviews Vol. , Issue 84j [page]

1984 Mathematical Reviews  
and other complexity problems may be independent of familiar formal mathematical systems, such as Peano arithmetic or even set theory.  ...  Caporaso, Salvatore; Pani, Giovanni 84j:03109 Undecidability vs. transfinite induction for the consistency of hyperarithmetical sets. Arch. Math. Logik Grundlag. 22 (1982), no. 1-2, 19-26.  ... 

Predicativity through transfinite reflection [article]

Andrés Cordon Franco, David Fernández Duque, Joost J. Joosten and Félix Lara Martín
2015 arXiv   pre-print
Peano Arithmetic is known to be provably equivalent to reflection over Elementary Arithmetic.  ...  Note that neither proof uses arithmetical comprehension but rather proceeds by transfinite induction. Clearly, one can bound the amount of transfinite induction by λ.  ...  This relation between reflection and a system of arithmetic can be extended to fragments of Peano arithmetic.  ... 
arXiv:1412.5521v2 fatcat:6p3yf4pfojdwtn2dipsd6uikvq

Arithmetic, Infinite Trees, and Second-order Subsystems: Notes and Observations [article]

David M. Cerna
2018 arXiv   pre-print
Work in progress concerning alternative formalizations of arithmetic.  ...  It is well known that Peano Arithmetic extended by the Axiom ACA 0 is a conservative extension of Peano arithmetic.  ...  Such an arithmetic has a strong induction schema which can prove the statement: FSA |= ∀h(h ∈ F 2 → ∃f (f ∈ F 1 ∧ ∀k(k ∈ N → {f | k} = {h | k} : k : k ))) This statement easily implies transfinite induction  ... 
arXiv:1610.04643v2 fatcat:7cgvjqvcdnekxbuuay5vjmycq4

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

J. J. Joosten
2012 arXiv   pre-print
In this paper we give an overview of an essential part of a Pi^0_1 ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev.  ...  This analysis is mainly performed within the polymodal provability logic GLP. We reflect on ways of extending this analysis beyond PA.  ...  of Peano Arithmetic (PA).  ... 
arXiv:1212.2395v1 fatcat:atuc7o3babgr3jq2ysq26wd3yq


2017 Journal of Symbolic Logic (JSL)  
Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction.  ...  Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ.  ...  The theory IΣ n is Peano arithmetic with induction restricted to Σ n -formulas.  ... 
doi:10.1017/jsl.2017.30 fatcat:ed5oxcysbnc7jljqry65qw4bva

Hydra Games and Tree Ordinals [chapter]

Ariya Isihara
2007 Lecture Notes in Computer Science  
Hydra games were introduced by Kirby and Paris, for the formulation of a result which is independent from Peano arithmetic but depends on the transfinite structure of 0.  ...  The termination of hydra games cannot be proved within Peano arithmetic, but under the assumption that the ordinal 0 is well-ordered.  ...  Transfinite induction on [ [nf (t)] ]. We suppose that [ [nf (s)] ] < [ [nf (t)]] implies s ∈ P . Case analysis by Lemma 19.  ... 
doi:10.1007/978-3-540-73445-1_17 fatcat:nagsqs427vgjzbw4qmrgssg5pa


2022 Bulletin of Symbolic Logic  
1 n (ε0) − denotes the schema of transfinite induction up to ε0 for Π 1 n formulas without set parameters.  ...  The main result is that for any second order arithmetic theory T0 extending RCA0 and axiomatizable by a Π 1 k+2 sentence, and for any n ≥ k + 1, , where T is T0 augmented with full induction, and TI Π  ...  It is well known that RCA 0 is a conservative extension of IΣ 1 , that is, Peano arithmetic PA with induction restricted to Σ 1 formulas.  ... 
doi:10.1017/bsl.2022.23 fatcat:ojlgdpuea5hzhlx7evl35o26lq

Are Induction and Well-Ordering Equivalent?

Lars–Daniel Öhman
2019 The Mathematical intelligencer  
by Peano's axioms is a well-ordering (p. 31), and transfinite induction is treated properly (pp. 53-54).  ...  Here, it could also be that what is intended is that transfinite induction in the case of the natural numbers coincides with ordinary induction.  ... 
doi:10.1007/s00283-019-09898-4 fatcat:cgs3o4bqjrfwljylpkiahu4ag4

Fast growing functions based on Ramsey theorems

H.J. Prömel, W. Thumser
1991 Discrete Mathematics  
Introduction Giidel's paper (1931) on formally undecidable propositions of first order Peano Arithmetic showed that any recursive axiomatic system which contains the axioms of Peano Arithmetic still admits  ...  For the reader who is not used to work in Peano Arithmetic we mention that (for statements about natural numbers) Peano Arithmetic is equivalent to the result of replacing the axiom of infinity by its  ... 
doi:10.1016/0012-365x(91)90346-4 fatcat:elwd3ybacjflfo66sjpolbmiua


2019 Bulletin of Symbolic Logic  
In 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to ε 0 , which was considered a finitistically acceptable procedure by both Gentzen and Paul  ...  This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays' finitary interpretation of transfinite induction.  ...  , and indeed, even beyond Peano Arithmetic.  ... 
doi:10.1017/bsl.2018.91 fatcat:aysfqpegergxfirhqrzl2c52a4

Some theories with positive induction of ordinal strength φω0

Gerhard Jäger, Thomas Strahm
1996 Journal of Symbolic Logic (JSL)  
forms of positive induction, and (iii) a subtheory of Peano arithmetic with ordinals in which induction on the natural numbers is restricted to formulas which are Σ in the ordinals.  ...  This paper deals with: (i) the theory which results from by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON(μ) plus various  ...  Peano arithmetic with ordinals and positive induction.  ... 
doi:10.2307/2275787 fatcat:jaqapknwzfd6zepithyc6jkjha

Poincaré's conjecture proved by G. Perelman by the isomorphism of Minkowski space and the separable complex Hilbert space

Васил Пенчев
2019 Figshare  
Then (8) the axiom of induction in Peano arithmetic should be replaced by transfinite induction correspondingly to (4) above, and (9) the statistical ensemble of well-orderings (as after measurement in  ...  Indeed, the axiom of induction in Peano arithmetic does not admit infinite natural numbers 3 .  ... 
doi:10.6084/m9.figshare.7901405.v1 fatcat:aahxhbb3dbc35fklnikvfnci7y

On Non-Standard Models of Peano Arithmetic and Tennenbaum's Theorem [article]

Samuel Reid
2013 arXiv   pre-print
and the properties that non-standard models of Peano arithmetic have.  ...  In the study of formalized theories of arithmetic, it is only natural to consider the extension from the standard model of Peano arithmetic, 〈N,+,×,≤,0,1 〉, to non-standard models of arithmetic.  ...  One such axiomatic system, known as Peano arithmetic, was proposed by the Italian mathematician Giuseppe Peano in his 1889, "Arithmetices principia, nova methodo exposita".  ... 
arXiv:1311.6375v1 fatcat:lunor2wjinhufkr2z563uouob4
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