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Injecting uniformities into Peano arithmetic

Fernando Ferreira
2009 Annals of Pure and Applied Logic  
We present a functional interpretation of Peano arithmetic that uses Gödel's computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic.  ...  As a consequence, some uniform boundedness principles (not necessarily set-theoretically true) are interpreted while maintaining unmoved the Π 0 2 -sentences of arithmetic.  ...  First, Peano arithmetic is interpreted into Heyting arithmetic by a negative translation.  ... 
doi:10.1016/j.apal.2008.09.004 fatcat:s7iktzew2bdavkaq6umnhjqrh4

LOGICISM, INTERPRETABILITY, AND KNOWLEDGE OF ARITHMETIC

SEAN WALSH
2014 The Review of Symbolic Logic  
Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical  ...  A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge.  ...  A map f : F → G is a bijection if it is injective and surjective.  ... 
doi:10.1017/s1755020313000397 fatcat:sa46d557ovaazbrgrrfxcufh3q

Reflection using the derivability conditions [chapter]

Matthews Sean, Alex Simpson
2017 Logic and algebra  
We extend arithmetic with a new predicate, Pr, giving axioms for Pr based on rst-order versions of L ob's derivability conditions.  ...  We hoped that the addition of a re ection schema mentioning Pr would then give a non-conservative extension of the original arithmetic theory. The paper investigates this possibility.  ...  Thus when we refer to Peano Arithmetic (PA) we mean a de nitional extension in L of the usual Peano Arithmetic (which is in the language of elementary arithmetic).  ... 
doi:10.1201/9780203748671-28 fatcat:u46e5cbtgbajbe4frdbfitopiq

Page 7537 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews  
Let N be a model of Peano arithmetic, and K be an initial segment of N closed under exponentiation. If a € N is definable (over @) in the pair (N, K) then a is definable in N over some b ¢ K.  ...  Let M be a countable recursively saturated model of Peano arithmetic and G = Aut(M ). Question |. (J. Schmerl) Is G elementarily equivalent to Aut(Q,<), where (Q,<) is the ordering of the rationals?  ... 

Declarative modeling of finite mathematics

Paul Tarau
2010 Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming - PPDP '10  
Conversely, arithmetic implementations of pairs, powersets, von Neumann ordinals shade new light on the bi-interpretability between Peano arithmetic and a theory of hereditarily finite sets.  ...  The main contribution of the paper is a fully constructive unification of paradigms -a chain of type classes does it all: Peano arithmetic, sets, sequences, binary trees, bitstrings.  ...  Peano arithmetic It is important to observe at this point that Peano arithmetic is also an instance of the class Polymath i.e. that the class can be used to derive an "axiomatization" for Peano arithmetic  ... 
doi:10.1145/1836089.1836107 dblp:conf/ppdp/Tarau10 fatcat:fyailg6kbrcdzocoqdt6umnvvu

Poor triviality and the sameness of Grothendieck semirings

Vinicius Cifú Lopes
2012 Proceedings of the American Mathematical Society  
Presburger arithmetic and first-order Peano arithmetic do not have poor triviality because R = { (x, y) ∈ N 2 | 0 y x } has fibers of any finite cardinality. 2.6.  ...  For example, models of first-order Peano arithmetic have trivial ring, but we will note that they do not have poor triviality.)  ... 
doi:10.1090/s0002-9939-2011-10977-8 fatcat:envl4xyj4jc65fmflzxxl2fale

A Most Artistic Package of a Jumble of Ideas

Fernando Ferreira
2008 Dialectica  
We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the "Dialectica" interpretation, discuss the extra principles that the interpretation validates, and comment  ...  An interpretation that directly injects uniformities into Peano arithmetic was recently defined in [Fer07] .  ...  VII), in injecting uniformities into mathematics.  ... 
doi:10.1111/j.1746-8361.2008.01134.x fatcat:6qllsb6xevgerlnxqetogisypy

Comparing Peano arithmetic, Basic Law V, and Hume's Principle

Sean Walsh
2012 Annals of Pure and Applied Logic  
The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic  ...  The main results of this paper are: (i) there is a consistent extension of the hyperarithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, and  ...  Introduction Second-order Peano arithmetic and its subsystems have been studied for many decades by mathematical logicians (cf.  ... 
doi:10.1016/j.apal.2011.12.016 fatcat:rtqyx3puabfkrcwej7pb4g34om

Unprovability in Mathematics: A First Course on Ordinal Analysis [article]

Anton Freund
2022 arXiv   pre-print
arithmetic (note that much stronger results of this type are due to Harvey Friedman).  ...  Our selection of topics is guided by the aim to give a complete and direct proof of a mathematical independence result: Kruskal's theorem for binary trees is unprovable in conservative extensions of Peano  ...  It may be interesting to observe that injectivity is automatic: Exercise 6.7. Show that any order reflecting f : P → Q between partial orders is injective.  ... 
arXiv:2109.06258v2 fatcat:oeutt7fqxvhudj7uuk6arnzbty

Page 4349 of Mathematical Reviews Vol. , Issue 90H [page]

1990 Mathematical Reviews  
Franco de Oliveira, L’arithmétique de Peano avec le prédicat “standard” [Peano arithmetic with the predicate standard] (pp. 331-341); Labib Haddad, Condorcet et les ultrafil- tres [Condorcet and ultrafilters  ...  Solovay, Injecting inconsistencies into models of PA (pp. 101- 132); Shi Qiang Wang, Inductive rings and fields (pp. 133-137); Mariko Yasugi, The machinery of consistency proofs (pp. 139- 152).  ... 

Weak arithmetical interpretations for the Logic of Proofs

Roman Kuznets, Thomas Studer
2016 Logic Journal of the IGPL  
Artemov established an arithmetical interpretation for the Logics of Proofs LP CS , which yields a classical provability semantics for the modal logic S4.  ...  In this paper, we remove this restriction by introducing weak arithmetical interpretations that are sound and complete for a wide class of constant specifications, including infinite ones.  ...  Peano Arithmetic PA is given in the language L PA .  ... 
doi:10.1093/jigpal/jzw002 fatcat:ce6inwoy7ra2xfn66cfmpxvnoa

The seven virtues of simple type theory

William M. Farmer
2008 Journal of Applied Logic  
It recommends that simple type theory be incorporated into introductory logic courses offered by mathematics departments.  ...  ., that Peano Arithmetic is categorical.  ...  Independently of Peano, R. Dedekind developed in [8] a theory of natural number arithmetic very similar to Peano Arithmetic.  ... 
doi:10.1016/j.jal.2007.11.001 fatcat:6sjnappxffbyrd73rdrsnpo46e

A new "feasible" arithmetic

Stephen Bellantoni, Martin Hofmann
2002 Journal of Symbolic Logic (JSL)  
Informally, one understands ⃞∝ as "∝ is feasibly demonstrable". differs from a system that is as powerful as Peano Arithmetic only by the restriction of induction to ontic (i.e., ⃞-free) formulas.  ...  A classical quantified modal logic is used to define a "feasible" arithmetic whose provably total functions are exactly the polynomial-time computable functions.  ...  Most cases are standard; we only treat those which significantly differ from the case of Peano arithmetic.  ... 
doi:10.2178/jsl/1190150032 fatcat:xni7n64gmrbgzbxfufag5oasoy

Maximal sets and fragments of Peano arithmetic

C.T. Chong
1989 Nagoya mathematical journal  
arithmetic.  ...  It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory?  ...  There is a model / of P~ + IΣ Q + ~^BΣ 1 with a Σ 2 injection p from f into an infinite subset of Jί. Proof. Let Jί be a nonstandard model of full Peano arithmetic.  ... 
doi:10.1017/s0027763000001604 fatcat:ja57xpwmfrgjlkg6zs3ak6zvoy

The Empty Set, The Singleton, and the Ordered Pair

Akihiro Kanamori
2003 Bulletin of Symbolic Logic  
surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano  ...  In his better known An Investigation into the Laws of Thought [1854] he had "signs" representing "classes", and incorporating the arithmetical property of 0 that 0 · y = 0 for every y, assigned [1854  ...  of "=" and moreover emphasized the difference between "0" and " 0" for arithmetic.  ... 
doi:10.2178/bsl/1058448674 fatcat:k4hxto6nobgcnfpmr6vbcg3cce
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