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Injecting uniformities into Peano arithmetic

2009
*
Annals of Pure and Applied Logic
*

We present a functional interpretation of

doi:10.1016/j.apal.2008.09.004
fatcat:s7iktzew2bdavkaq6umnhjqrh4
*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. ...##
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LOGICISM, INTERPRETABILITY, AND KNOWLEDGE OF ARITHMETIC

2014
*
The Review of Symbolic Logic
*

Here an implementation of this idea is considered that holds that knowledge of

doi:10.1017/s1755020313000397
fatcat:sa46d557ovaazbrgrrfxcufh3q
*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. ...##
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Reflection using the derivability conditions
[chapter]

2017
*
Logic and algebra
*

We extend

doi:10.1201/9780203748671-28
fatcat:u46e5cbtgbajbe4frdbfitopiq
*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*). ...##
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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? ...##
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Declarative modeling of finite mathematics

2010
*
Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming - PPDP '10
*

Conversely,

doi:10.1145/1836089.1836107
dblp:conf/ppdp/Tarau10
fatcat:fyailg6kbrcdzocoqdt6umnvvu
*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*...##
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Poor triviality and the sameness of Grothendieck semirings

2012
*
Proceedings of the American Mathematical Society
*

Presburger

doi:10.1090/s0002-9939-2011-10977-8
fatcat:envl4xyj4jc65fmflzxxl2fale
*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.) ...##
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A Most Artistic Package of a Jumble of Ideas

2008
*
Dialectica
*

We also make some observations concerning Gödel's recasting of intuitionistic

doi:10.1111/j.1746-8361.2008.01134.x
fatcat:6qllsb6xevgerlnxqetogisypy
*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. ...##
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Comparing Peano arithmetic, Basic Law V, and Hume's Principle

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

doi:10.1016/j.apal.2011.12.016
fatcat:rtqyx3puabfkrcwej7pb4g34om
*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. ...##
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Unprovability in Mathematics: A First Course on Ordinal Analysis
[article]

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*. ...

##
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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). ...##
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Weak arithmetical interpretations for the Logic of Proofs

2016
*
Logic Journal of the IGPL
*

Artemov established an

doi:10.1093/jigpal/jzw002
fatcat:ce6inwoy7ra2xfn66cfmpxvnoa
*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 . ...##
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The seven virtues of simple type theory

2008
*
Journal of Applied Logic
*

It recommends that simple type theory be incorporated

doi:10.1016/j.jal.2007.11.001
fatcat:6sjnappxffbyrd73rdrsnpo46e
*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*. ...##
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A new "feasible" arithmetic

2002
*
Journal of Symbolic Logic (JSL)
*

Informally, one understands ⃞∝ as "∝ is feasibly demonstrable". differs from a system that is as powerful as

doi:10.2178/jsl/1190150032
fatcat:xni7n64gmrbgzbxfufag5oasoy
*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*. ...##
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Maximal sets and fragments of Peano arithmetic

1989
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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*. ...

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The Empty Set, The Singleton, and the Ordered Pair

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

doi:10.2178/bsl/1058448674
fatcat:k4hxto6nobgcnfpmr6vbcg3cce
*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*. ...
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