## Non-Standard Models of Arithmetic: a Philosophical and Historical perspective (2010)

### BibTeX

@MISC{Giorgio10non-standardmodels,

author = {Nicola Di Giorgio},

title = {Non-Standard Models of Arithmetic: a Philosophical and Historical perspective },

year = {2010}

}

### OpenURL

### Abstract

### Citations

569 | Model Theory
- Hodges
- 1993
(Show Context)
Citation Context ...ars a certain resemblance to the method used above. I am indebted to Professor Church for bringing this paper to my attention. (Added February 14, 1950.)” [henkin50, p.90, fn.12]. See in this respect =-=[hodges09]-=- 703.3. COMPACTNESS AND THE MODERN PROOF OF THE EXISTENCE OF NMOA Theorem 3.4. PA has non-standard models Proof. Let N be the standard model of PA, i.e. N = (N, 0, S, +, ·) where N is the set of natu... |

419 |
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I.Monatshefte für Mathematik und Physik 38
- Gödel
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(Show Context)
Citation Context ...ondition concerning the notion of chain requires second order logic to be expressible. To be more precise, it amounts to the following second order formula: 96 ∀X[(X0 ∧ ∀x(Xx → XS(x))) → ∀xXx] 95 See =-=[godel31]-=- 96 See appendix for further details on second-order logic. 562.7. TO SUM UP The way this whole line of research was born suggests that the most natural framework for foundational inquiry is the seco... |

233 | Non-standard analysis - Robinson - 1996 |

201 |
Computability and Logic
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- 2003
(Show Context)
Citation Context ...-standard models have been introduced by Skolem, in a series of papers from 1922 to 1934, in two cases: set theory and arithmetic. [gaifman03, p. 2] In other introductory texts, e.g. [vandalen04] and =-=[boolos07]-=-, inasmuch as the technical facet supersedes historical details on Skolem’s endeavour, Van Dalen and Boolos simply put the existence of non-standard models as a consequence of Gödel’s Compactness theo... |

126 |
Logic and Structure
- Dalen
- 1983
(Show Context)
Citation Context ...ges01, p. 70] Non-standard models have been introduced by Skolem, in a series of papers from 1922 to 1934, in two cases: set theory and arithmetic. [gaifman03, p. 2] In other introductory texts, e.g. =-=[vandalen04]-=- and [boolos07], inasmuch as the technical facet supersedes historical details on Skolem’s endeavour, Van Dalen and Boolos simply put the existence of non-standard models as a consequence of Gödel’s C... |

51 |
Foundations without Foundationalism. A Case for Second-Order Logic
- Shapiro
- 1991
(Show Context)
Citation Context ...e of which logic to employ should not be based on the different meta-properties that each logic has per se, but rather on the foundational purpose in mathematics that we want to pursue. 161 Commonly, =-=[shapiro91]-=- represents the landmark in this debate. 162 [shapiro85, p.715] 853.7. DESCRIPTIVE VS DEDUCTIVE USE: 1ST VS 2ND ORDER LOGIC Meta-properties of first and second order logic are recapitulated in the ta... |

47 |
What numbers could not be
- Benacerraf
- 1965
(Show Context)
Citation Context ...e impact that Tennenbaum’s Theorem along the lines proposed in [halbach-horsten05]. 4.3.1 Tennenbaum’s Theorem and Structuralism The structuralist account in philosophy of mathematics was sketched in =-=[benacerraf65]-=-. Structuralism comes out as an attempt to bring philosophical explanations more in line with actual mathematical practice. In particular, Benacerraf’s structuralism is inspired by the consideration t... |

45 | 1889, Arithmetices principia nova methodo exposita - Peano |

40 |
Elementary predicate logic
- Hodges
- 2001
(Show Context)
Citation Context ...ion symbol f and each relation symbol R, which belong respectively to the sets fun(L) and rel(L), are assumed to be labelled somehow to indicate 191 The following exposition is a blend of [oosten10], =-=[hodges01]-=-, [vandalen04] and [awodey02] 110that they are an n-place function constant and an m-place predicate constant (for some positive integer n and m), and we call it the arity of f or of R. Definition A.... |

25 |
Heijenoort, From Frege to Gödel. A Source Book in Mathematical Logic
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- 1967
(Show Context)
Citation Context .... In particular, his translation undermines the evidence for Dedekind’s construction of the non-standard models. Let us compare Bauer-Menglberg’s translation, which we have quoted above, published in =-=[vanheijenoort67]-=- All these facts would hold also for every system S that, besides the number sequence N, contained a system T, of arbitrary additional elements t, to which the mapping ϕ could always be extended while... |

21 |
Models of Peano Arithmetic. Oxford Logic Guides, 15. Oxford Science Publications
- Kaye
- 1991
(Show Context)
Citation Context ...y two fundamental references on the non-standard models, so-called ”bibles” by the insiders, that do justice to what should be regarded, by this time, as an important and promising field of research: =-=[kaye91]-=- and the more recent [kossak06]. 2 mathematically oriented. On top of that, we should note that these two monographs are strongly At the end of the day, it would then seem that the concept of non-stan... |

21 | A Course in Model Theory. An Introduction to Contemporary Mathematical Logic - Poizat |

19 |
Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics
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- 1999
(Show Context)
Citation Context ...] for comparison with Frege, and [reck03] for similarities with Kronecker 37 See the table on page 31 38 From now on, the word system should be taken in Dedekind’s sense, i.e. a mathematical set. See =-=[ferreiros07]-=- 39 Roughly speaking, a simply infinite system represents a mathematical structure that has the same properties of the structure of the series of the natural numbers. 232.2. LETTER TO KEFERSTEIN Roug... |

19 |
Arithmetical extensions of relational systems
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- 1956
(Show Context)
Citation Context ... given his name: Legend has it that Thoralf Skolem, up until the end of his life, was scandalized by the association of his name to a result of this type, which he con159 [read97, p.83] 160 Ed. note: =-=[tarski57]-=- 843.7. DESCRIPTIVE VS DEDUCTIVE USE: 1ST VS 2ND ORDER LOGIC sidered an absurdity, nondenumerable sets being, for him, fictions without real existence. [poizat00, pp. 53–54]. 3.7 Descriptive vs deduc... |

19 | Second-order logic and foundations of mathematics
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(Show Context)
Citation Context ...6 See page 40. 167 E.g., see Wang’s passage on page 81. 168 That is [skolem33] and [skolem34] 169 [godel86, p. 376] 170 E.g., see the claim of the theorem 3.5.2. 171 Ed. note: [henkin50] 172 E.g. see =-=[vaananen01]-=- and [odifreddi03, p. 274] 173 See section 3.3 above. 924.1. AN EXEGETICAL AND HISTORICAL CLARIFICATION standard semantics” has the same meta-properties as first-order logic. 174 The bivalent use of ... |

15 | Thinking about Mathematics - Shapiro - 2000 |

11 |
The Structure of Models of Peano Arithmetic
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- 2007
(Show Context)
Citation Context ...on the non-standard models, so-called ”bibles” by the insiders, that do justice to what should be regarded, by this time, as an important and promising field of research: [kaye91] and the more recent =-=[kossak06]-=-. 2 mathematically oriented. On top of that, we should note that these two monographs are strongly At the end of the day, it would then seem that the concept of non-standard models has little to contr... |

10 |
Lectures on nonstandard models of arithmetic
- Smorynski, Smorynski
- 1984
(Show Context)
Citation Context ...as little to contribute to philosophical debates. Roughly speaking, the two mathematical studies along with the meagre logical treatment exhaust the literature on non-standard models. An exception is =-=[smorynski84]-=- which was indeed a forerunner attempt to give a historical account of non-standard models. Before the publishing of [kaye91] in 1991, ”the only, and not easily available, source was the excellent not... |

9 | Meditationes de prima philosophia - Descartes |

9 |
Second-order languages and mathematical practice
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- 1985
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Citation Context ...r one comes at a price. Crucially, the understanding of second-order consequence takes for granted an a priori un186 See [halbach-horsten05, pp. 175–176] 187 See [halbach-horsten05, p. 176] 188 E.g., =-=[shapiro85]-=- and [shapiro91] 1034.3. A PHILOSOPHICAL REVIVAL? derstanding of the notion of power set. In fact, the power set is a more problematic than the notion of natural number itself. 189 Despite the feasib... |

7 | Philosophy of Mathematics; Selected Readings. Second Edition - Benacerraf, Putnam - 1983 |

7 |
Structures and structuralism in contemporary philosophy of mathematics. Synthese
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- 2000
(Show Context)
Citation Context ...erstein addresses strong criticism on Dedekind’s tenets. What is at issue is the choice of Dedekind’s assumptions and the way he had developed them throughout the paper. 34 See [tait96], [reck03] and =-=[reck00]-=- 35 See [wang57, pp. 156–157] 36 See [tait96] for comparison with Frege, and [reck03] for similarities with Kronecker 37 See the table on page 31 38 From now on, the word system should be taken in Ded... |

7 | Frege versus Cantor and Dedekind: On the concept of number
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- 1997
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Citation Context ...rs”. In particular, Keferstein addresses strong criticism on Dedekind’s tenets. What is at issue is the choice of Dedekind’s assumptions and the way he had developed them throughout the paper. 34 See =-=[tait96]-=-, [reck03] and [reck00] 35 See [wang57, pp. 156–157] 36 See [tait96] for comparison with Frege, and [reck03] for similarities with Kronecker 37 See the table on page 31 38 From now on, the word system... |

7 |
Non-standard models for formal logics
- Rosser, Wang
- 1950
(Show Context)
Citation Context ...allow only standard interpretation of quantifiers you get only the standard model. Conversely, if you allow non-standard interpretation of quantifiers, then you get non-standard models. 177 Ed. note: =-=[wang-rosser50]-=- 178 Ed. note: [wang-rosser50] 944.1. AN EXEGETICAL AND HISTORICAL CLARIFICATION the equality relation in the logic or the part of the model which represents the positive integers is not well-ordered... |

6 |
Dedekind’s structuralism: an interpretation and partial defense
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(Show Context)
Citation Context ...rticular, Keferstein addresses strong criticism on Dedekind’s tenets. What is at issue is the choice of Dedekind’s assumptions and the way he had developed them throughout the paper. 34 See [tait96], =-=[reck03]-=- and [reck00] 35 See [wang57, pp. 156–157] 36 See [tait96] for comparison with Frege, and [reck03] for similarities with Kronecker 37 See the table on page 31 38 From now on, the word system should be... |

5 |
Completeness and Categoricity. Part I: Nineteenth-Century Axiomatics to Twentieth-Century Metalogic”, History and Philosophy of Logic 23
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- 2002
(Show Context)
Citation Context ...ject of the science of numbers or arithmetic. [dedekind01, p. 34] The whole discussion we have put forward about the sixth point of Dedekind’s letter seems to be in flat contradiction with a claim in =-=[awodey02]-=-: At the same time, Dedekind clearly intends various different systems to fall under the concept ‘simply infinite system’. He even considers systems that satisfy only some of the four clauses in it bu... |

5 |
1970]: 'A survey of Skolem's work in logic
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Citation Context ...Since the original [skolem33] is in German, this is the English translation presented by Wang which hardly differs from the English translation of Gödel’s review. 138 Skolem 132 [godel86, p. 376] 133 =-=[wang70]-=- 134 [godel86, pp. 376–379] 135 [smorynski84] 136 See [godel34, p.379] 137 Wang’s translation on [wang70, p. 41] 138 [godel34, p. 379] 723.5. SKOLEM’S PROOF OF THE EXISTENCE OF NMOA defines N to be t... |

4 | Skolem’s Paradox
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Citation Context ...than, Skolem used to adopt long periphrases instead of special terms. 110 Today we actually know that there are other types of non-standard models of set theory other than the countable ones. 111 See =-=[bays09]-=- 112 See the discussion in section 3.6 643.1. SKOLEM AND NON-STANDARD MODELS Toward the end of the 1929 paper 113 Skolem expressed some doubts as to the complete axiomatizability of mathematical conc... |

4 |
The nature and the meaning of numbers. In Essays on the theory of numbers. the open court publishing company
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Citation Context ...retical techniques. In this respect, we find a clearly stated train of definitions and theorems that hardly differs from modern presentation in set theory 33 . 32 We will cite the English translation =-=[dedekind01]-=- 33 For details see [reck03, pp. 374–375] 222.1. DEDEKIND AND THE BIRTH OF NON-STANDARD MODELS • Secondly, Dedekind offers a brand-new philosophical standpoint concerning mathematics and its subject ... |

4 | Sul concetto di numero, Rivista di mateincztica - Peano |

3 | Non-standard models in a broader perspective - Gaifman - 2003 |

3 |
The Completeness of Formal Systems
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- 1947
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Citation Context ...ion of the standard model N , rather than denoting them as an independent class of models. 170 The standard/non-standard terminology was adopted for the first time by Henkin in his Ph.D. dissertation =-=[henkin47]-=-. An extract of his dissertation was published in 1950 in The Journal of Symbolic Logic with the title ”Completeness in the Theory of Types” 171 . This paper was a landmark in the history of logic for... |

3 | Tennenbaum’s Theorem for Models of Arithmetic - Kaye - 2011 |

3 |
Modelle der Arithmetik
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Citation Context ...xtension M∗ in which “all new elements follow all old”. In this case, Skolem’s notion of “definability” presented earlier comes out as essential for 152 N N represents the Cartesian product ∏ 153 See =-=[macdowell-specker61]-=- i∈N Ni. 793.6. SKOLEM’S PHILOSOPHY AND THE PROOF OF THE EXISTENCE OF NMOA MacDowell’s and Specker’s argument. In fact their proof is related to Skolem’s insofar as MacDowell and Specker construct M ... |

3 |
Peano’s axioms and models of arithmetic
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Citation Context ...lem in Memoriam”: 108 For a definition of Skolem functions, see [wang70, p.19–22] 109 Indeed, Skolem never used the term “non-standard” for these models except for the last paper on the topic, namely =-=[skolem55]-=-. Before than, Skolem used to adopt long periphrases instead of special terms. 110 Today we actually know that there are other types of non-standard models of set theory other than the countable ones.... |

2 |
The Algebra of Logic tradition
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Citation Context ... this fact and said that Gödel had discovered the theorem independently and given it an attractive treatment. [wang96, p.122] 156 See the influence of the algebraic tradition on Skolem’s formation in =-=[burris09]-=- and his isolated condition from the research community for a long period of his life in [fenstad70, pp. 9–16] 157 [wang96, p.119] 813.6. SKOLEM’S PHILOSOPHY AND THE PROOF OF THE EXISTENCE OF NMOA Ho... |

2 |
Aristotle on causality
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Citation Context ...s advice, the father of the child. iv) The final cause: “the end, that for the sake of which a thing is done”, e.g., health is the end of walking, losing weight, purging, drugs, and surgical tools. ” =-=[falcon08]-=- 100 See section 3.7.1 below. 59Chapter 3 The philosophical phase - Skolem and the proof of the existence of NMoA In the previous chapter we have provided some arguments in favour of the fact that De... |

2 |
Is there completeness in mathematics after Gödel
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Citation Context ... See theorem 2.5.0.2 on page 48. Once again, according to the distinction drew in section 1.3, here completeness refers to a meta-property pertaining to logic and we call it “exhaustiveness”. 128 See =-=[hintikka89]-=- 129 See [barendregt10] 130 Ed. note: [godel34] 693.3. COMPACTNESS AND THE MODERN PROOF OF THE EXISTENCE OF NMOA either: also does not mention this fact, rather observing that the failure of categori... |

2 | The skeleton in Frege’s cupboard: The standard versus nonstandard distinction’, The Journal of Philosophy, 89, 290–315. Dedekind, and the Origins of Logicism 265 Downloaded by - Hintikka, Sandu - 1992 |

2 | Completeness and Categoricity: Frege, Gödel and Model Theory. History and Philosophy of Logic - Read - 1997 |

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Gödel’s incompleteness theorems — lecture notes. http: //www.cs.ru.nl/˜henk/courses.html
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Citation Context ...on page 48. Once again, according to the distinction drew in section 1.3, here completeness refers to a meta-property pertaining to logic and we call it “exhaustiveness”. 128 See [hintikka89] 129 See =-=[barendregt10]-=- 130 Ed. note: [godel34] 693.3. COMPACTNESS AND THE MODERN PROOF OF THE EXISTENCE OF NMOA either: also does not mention this fact, rather observing that the failure of categoricity for arithmetic fol... |

1 |
Recantation or any old omega-sequence would do after all
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Citation Context ...rtant methods. 186 To begin with, in [benacerraf65] two conditions are established: on the one hand the structure is an ω-sequence, and on the other hand the ordering of the elements is recursive. In =-=[benacerraf96]-=-, Benacerraf drops the requirement of recursiveness and leaves only the condition concerning the order type. After all, to assume that each model has to be an ω-sequence is surely sufficient to rule o... |

1 | Schemata: The concept of schema in the history of logic
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Citation Context ...2, p. 5] 10 See [dutilh-novaes10] for an historical account of the notion of formality. 11 Note that here the concept of scheme is not related to the one of induction scheme presented earlier. 12 See =-=[corcoran06]-=- for a historical account of the concept of “schema” and “scheme”. 41.3. HINTIKKA AND THE TWO USES OF LOGIC IN MATHEMATICS 1.3 Hintikka and the two uses of logic in mathematics A formal language prov... |

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1 |
Letter to keferstein (1890
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Citation Context ..., Dedekind clearly intends various different systems to fall under the concept ‘simply infinite system’. He even considers systems that satisfy only some of the four clauses in it but not others; see =-=[dedekind90]-=-, pp.100101. [awodey02, p. 7, fn. 21, emphasis mine] Apparently, Awodey and Reck fail to grasp what Dedekind made extremely clear. Dedekind points out that the three conditions α, γ and δ are not enou... |

1 |
Novaes. What does it mean to say that something is formal
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Citation Context ... of the properties that objects have as a components of a common structure. Hence, we will use formal axiomatics as an essential device to study mathematical structures. 9 See [awodey02, p. 5] 10 See =-=[dutilh-novaes10]-=- for an historical account of the notion of formality. 11 Note that here the concept of scheme is not related to the one of induction scheme presented earlier. 12 See [corcoran06] for a historical acc... |

1 |
Thoralf Albert Skolem in Memoriam
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(Show Context)
Citation Context ...dical finitary position also after Gödel’s result. But what do Skolem’s finitary methods exactly mean? To answer this question, Fenstad for example brings into consideration one of Skolem’s papers in =-=[fenstad70]-=-: His [Skolem’s] own preferences are better represented by the important 1923 paper. Here Skolem tries to build up elementary arithmetic without applying the unrestricted quantifiers “for all” and “th... |

1 | Teoria della fondazione epistemica - Giordani - 2002 |

1 |
Besprechung von ”Über die Unmöglichkeit einer vollstandigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems”. Zentralblatt für Mathematik und ihre Grenzgebiete
- Gödel
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(Show Context)
Citation Context ...lem puts forth this result in other three papers: [skolem34], [skolem41] and [skolem55]. Let us discuss them in turn. The 1933 and 1934 papers are both written in German and both reviewed by Gödel in =-=[godel34]-=- and [godel35] respectively. Gödel acknowledges the fact that the two papers are essentially one, as he reports in the review to [skolem34]: With respect to its result, this work is identical to the s... |

1 |
Besprechung von ”Über de Nichtcharakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Zentralblatt für Mathematik und ihre Grenzgebiete, 10:49
- Gödel
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(Show Context)
Citation Context ... this result in other three papers: [skolem34], [skolem41] and [skolem55]. Let us discuss them in turn. The 1933 and 1934 papers are both written in German and both reviewed by Gödel in [godel34] and =-=[godel35]-=- respectively. Gödel acknowledges the fact that the two papers are essentially one, as he reports in the review to [skolem34]: With respect to its result, this work is identical to the same author’s 1... |