## Number theory and elementary arithmetic (2003)

Venue: | Philosophia Mathematica |

Citations: | 17 - 5 self |

### BibTeX

@ARTICLE{Avigad03numbertheory,

author = {Jeremy Avigad},

title = {Number theory and elementary arithmetic},

journal = {Philosophia Mathematica},

year = {2003},

volume = {11},

pages = {2003}

}

### OpenURL

### Abstract

Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1

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Citation Context ...spectively, Π1) if it is equivalent to the result of adding a single block 6 There are a number of equivalent characterizations of the Kalmar elementary functions, many of which are presented in Rose =-=[61]-=-. Rose credits the definition of the elementary functions to Kalmar (1943) and Csillag (1947). 7 The interpretation of set theory in even weaker fragments of arithmetic has been considered by Sazonov,... |

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Citation Context ...ism towards formalism, there is currently a good deal of interest in specifying formal deductive systems that represent the informal practice, and understanding their metamathematical properties. See =-=[9, 12, 80]-=-, and [13] for a recent survey. From the 1950’s onwards, Georg Kreisel urged the application of proof theoretic methods towards obtaining additional mathematical information from nonconstructive proof... |

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Citation Context ...h adds Σ1 induction to WKL∗ 0 , is conservative over primitive recursive arithmetic for Π2 sentences. Proof-theoretic proofs were later obtained by Sieg [68], who used cut elimination, and Kohlenbach =-=[39, 41]-=-, who used the Dialectica interpretation and extended the result to weaker theories (see also [6]). Leo Harrington was able to strengthen Friedman’s theorem by showing that WKL0 is conservative over R... |

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Citation Context ...can typically construct a closed term t and a proof, in EA, of the assertion ∃d ′ < t (“d ′ is a proof of ϕ in EA”). Then, using a partial truth predicate and Solovay’s method of shortening cuts (see =-=[29, 57]-=-) one can construct a short proof of the soundness of EA up to t (for formulae of complexity less than or equal to that of ϕ); and hence a short proof of ϕ. To be sure, derivations will get longer whe... |

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Citation Context ... Harvey Friedman and Jeffrey Paris have shown, independently and using different model-theoretic proofs, that one can add this principle to EA, without changing the Π2 consequences of the theory (see =-=[29, 35]-=-). 8 Wilfried Sieg [66] has used Gerhard Genzten’s method of cut elmination to obtain an effective translation from one theory to the other (see also [10], or [2] for a model-theoretic version of Sieg... |

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Citation Context ...nder recursive definability and containing a path through every infinite binary tree, is called a Scott set. Their importance to the study of models of arithmetic was first demonstrated by Dana Scott =-=[63]-=-, and was rediscovered and put to dramatic use by Friedman (see Kaye [35]). 10scharacters, that is, functions χ that map the group of natural numbers relatively prime to d homomorphically to complex r... |

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Citation Context ...h adds Σ1 induction to WKL∗ 0 , is conservative over primitive recursive arithmetic for Π2 sentences. Proof-theoretic proofs were later obtained by Sieg [68], who used cut elimination, and Kohlenbach =-=[39, 41]-=-, who used the Dialectica interpretation and extended the result to weaker theories (see also [6]). Leo Harrington was able to strengthen Friedman’s theorem by showing that WKL0 is conservative over R... |

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Citation Context ...ch has made important progress in the use of metamathematical methods to extract useful information and sharper results from nonconstructive proofs in numerical analysis and approximation theory. See =-=[41, 42, 37]-=- for an overview of the methods, and [40, 43] for some applications. Contemporary proof theory can be characterized broadly as the general study of deductive systems, mathematical or otherwise. By now... |

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Citation Context ...e ϕ is bounded, then there is a term t, not involving y, such that EA also proves ∀x ∃y < t ϕ(x, y). This second theorem is a special case of a more general theorem due to Rohit Parikh [54] (see also =-=[14]-=-), and implies that each ∆0-definable function of EA is bounded by a finite iterate of the exponential function. A first objection to the claim that a good deal of mathematics can be carried out in EA... |

29 |
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Citation Context ...on of Selberg’s result, as well as a slightly less elementary proof that uses infinite limits but avoids complex analysis, can be found in Melvyn Nathanson’s book, Elementary methods in number theory =-=[52]-=-. Georg Kreisel mentioned Dirichlet’s theorem in [47, Section 3.3] and [48, Section 3.2] (see also [50]) as an example of a proof amenable to his “unwinding” program, which was designed to extract use... |

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Citation Context ...however, there is a long tradition, from Weyl’s Das Kontinuum [81] and Hilbert and Bernays’ Grundlagen der Mathematik [32] through the work of Takeuti [77] and the school of Reverse Mathematics today =-=[71]-=-, of studying theories that are significantly weaker. The general feeling is that most “ordinary” mathematics can be carried out, formally, without using the full strength of ZFC . I should qualify th... |

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26 |
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Citation Context ...y Paris have shown, independently and using different model-theoretic proofs, that one can add this principle to EA, without changing the Π2 consequences of the theory (see [29, 35]). 8 Wilfried Sieg =-=[66]-=- has used Gerhard Genzten’s method of cut elmination to obtain an effective translation from one theory to the other (see also [10], or [2] for a model-theoretic version of Sieg’s proof). More recentl... |

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Citation Context ...many of the usual finitary techniques of number theory and combinatorics can not be carried out. See [56] and [18] for examples of what can be done in such theories. For 18 See, for example, Harrison =-=[31]-=-. 20san overview of the subject, as well as some of the interesting connections to the subject of computational complexity, see Krajícek [45], as well as [29, 59]. As far as the formalization of mathe... |

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Citation Context ...er XXI] and [52, Section 11.5] for notes and references.) On the other hand, Roth’s theorem is, famously, a Π3 assertion for which it is still open as to whether there is a computable bound; see e.g. =-=[49, 50]-=-. See also the discussion of the prime number theorem in footnote 12 below. 9 It is still open as to whether Σ1 collection is strictly stronger than ∆1 induction; see, for example, [29, 11]. 8sRCA of ... |

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19 |
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Citation Context ... of mathematics, see, for example, [22, 53, 69]. 19 Wilfried Sieg points out that the spirit of such reversals can be found as early as 1872 in Dedekind’s famous Stetigkeit und die irrationale Zahlen =-=[19]-=-, at the end of which Dedekind shows that his “continuity principle” is in fact equivalent to fundamental theorems of analysis like the least upper-bound principle. The same spirit can be found in the... |

19 | A quantitative version of a theorem due to Borwein-Reich-Shafrir
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Citation Context ...mathematical methods to extract useful information and sharper results from nonconstructive proofs in numerical analysis and approximation theory. See [41, 42, 37] for an overview of the methods, and =-=[40, 43]-=- for some applications. Contemporary proof theory can be characterized broadly as the general study of deductive systems, mathematical or otherwise. By now, a sprawling array of loosely affiliated dis... |

18 | A feasible theory for analysis
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(Show Context)
Citation Context ... dramatic use of analytic methods. In modern terms, to each value of d one assigns a group of Dirichlet a version of the conservation result for a theory of polynomial time computable arithmetic (see =-=[25, 24]-=-). There are also connections between weak König’s lemma and nonstandard arithmetic (see Avigad [3] and Tanaka [78]). See also Kohlenbach [38] for a discussion of uniform versions of weak König’s lemm... |

17 | Proof theory and computational analysis
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Citation Context ...s shown that various forms of Gödel’s Dialectica intepretation can also be used to obtain conservation results, yielding additional information. Indeed, as part of his proof mining program Kohlenbach =-=[37]-=- has developed many analytic notions—continuous functions, integration, transcendental functions like e x , sine and cosine—in theories that are even weaker, and has explored ways of eliminating vario... |

17 | Stetigkeit und irrationale Zahlen - Dedekind |

16 | Formalizing forcing arguments in subsystems of secondorder arithmetic
- Avigad
- 1996
(Show Context)
Citation Context ...ng weak König’s lemma does not change the Π 1 1 theorems of RCA ∗ 0 . Simpson and Smith used a model-theoretic argument to prove the conservation theorem for RCA ∗ 0 , but using the methods of Avigad =-=[4]-=- a direct interpretation of WKL ∗ 0 in RCA ∗ 0 can also be obtained. 10 10The unrestricted version of König’s lemma [44] asserts that every finitely branching tree with arbitrarily long paths has an i... |

16 | New effective moduli of uniqueness and uniform a–priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory. Numerical Functional Analysis and Optimization
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(Show Context)
Citation Context ...mathematical methods to extract useful information and sharper results from nonconstructive proofs in numerical analysis and approximation theory. See [41, 42, 37] for an overview of the methods, and =-=[40, 43]-=- for some applications. Contemporary proof theory can be characterized broadly as the general study of deductive systems, mathematical or otherwise. By now, a sprawling array of loosely affiliated dis... |

16 | 1978]: Two Applications of Logic to Mathematics - Takeuti |

15 |
1993] “Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable
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(Show Context)
Citation Context ...that much of ordinary mathematics takes place in Zermelo set theory; so much so that Quine judged objects unavailable in that theory to be “without ontological rights” (Quine [58], quoted in Feferman =-=[20]-=-). Without an axiom of infinity, we have to relinquish the set of natural numbers, and we have restricted separation and foundation in the axioms above; but otherwise, Zermelo set theory remains intac... |

14 | A mathematical incompleteness - Paris, Harrington - 1977 |

13 |
Eliminating definitions and Skolem functions in first-order logic
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- 2001
(Show Context)
Citation Context ... come up in practice. For example, most of the conservation results described in Section 2 can be obtained by direct interpretation, with a polynomial bound on the increase in length of proof. Avigad =-=[5]-=-, for example, provides an efficient way of eliminating symbols that are introduced in definitional extensions. Although translations that use cut elimination and normalization can lead to superexpone... |

13 | 1937]: ‘Die Widerspruchsfreiheit der allgemeinen Mengenlehre - Ackermann |

11 | Kreisel’s ‘Unwinding Program
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(Show Context)
Citation Context ...tic methods towards obtaining additional mathematical information from nonconstructive proofs in various branches of mathematics. His “unwinding program” met with mixed results; see the discussion in =-=[21, 50]-=- and other essays in [53]. But, under the rubric of “proof mining,” Ulrich Kohlenbach has made important progress in the use of metamathematical methods to extract useful information and sharper resul... |

11 | Does mathematics need new axioms
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(Show Context)
Citation Context ... statements, there are, for example, Goodstein’s theorem, the Paris-Harrington theorem, and Friedman’s finitary version of Kruskal’s theorem (see, for example, [55, 26] and Friedman’s contribution to =-=[23]-=-). But all of these were designed by logicians who were explicitly looking for independent statements of this sort. Restricting our attention to ordinary mathematical literature, then, there is some p... |

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11 | On bounded set theory
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(Show Context)
Citation Context ...edits the definition of the elementary functions to Kalmar (1943) and Csillag (1947). 7 The interpretation of set theory in even weaker fragments of arithmetic has been considered by Sazonov, e.g. in =-=[62]-=-. 7sof existential (respectively universal) quantifiers to a ∆0 formula. A formula is said to be ∆1 if it is provably equivalent to both Σ1 and Π1 formulae. Bearing in mind that ∆0 properties are fini... |

10 | On uniform weak König’s lemma
- Kohlenbach
- 2002
(Show Context)
Citation Context ... theory of polynomial time computable arithmetic (see [25, 24]). There are also connections between weak König’s lemma and nonstandard arithmetic (see Avigad [3] and Tanaka [78]). See also Kohlenbach =-=[38]-=- for a discussion of uniform versions of weak König’s lemma in a higher-order setting. An ω-model of WKL0 , i.e. a collection of sets closed under recursive definability and containing a path through ... |

9 | Finitism - Tait - 1981 |

8 |
Free-variable axiomatic foundations of infinitesimal analysis: afragment withfinitary consistency proof
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- 1995
(Show Context)
Citation Context ...an even extend higher-order conservative extensions of elementary arithmetic with “nonstandard” reasoning, i.e. reasoning involving nonstandard natural numbers as well as infinitesimal rationals (see =-=[3, 7, 17, 73]-=-). This provides yet another weak framework in which one can develop analytic notions in a natural way. 3 Case studies An arithmetic progression is a sequence of the form a, a + d, a + 2d, . . . where... |

8 |
Interpretability and fragments of arithmetic
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- 1993
(Show Context)
Citation Context ...ard Genzten’s method of cut elmination to obtain an effective translation from one theory to the other (see also [10], or [2] for a model-theoretic version of Sieg’s proof). More recently, Petr Hájek =-=[28]-=- has shown that one may even obtain this conservation result by a direct intepretation. The principle of Σ1 collection can be used to justify the principle of induction for ∆1 formulae, i.e. induction... |