Skolem is commonly considered as the father of non-standard models. Yet, as we showed, he has to share this credit with Dedekind. Unlike Dedekind’s, Skolem’s work takes place in a period which was characterised by a change in research goals and in the status of the field of mathematical logic. This chapter will be devoted to the main differences between Skolem’s and Dedekind’s attempt, to Skolem’s proof of the existence of NMoA for PA, to Skolem’s philosophical views and its influence on the development of his work, and finally we will go back to the relation between descriptive and deductive uses of logic in mathematics in the light of the first versus second order debate over the choice of the logical framework.
3.1
Skolem and non-standard models
In the 1930s, logic had reached its own autonomy, clear boundaries, importance as a subject matter, and thus its own tools and methods. In particular, mathematical logic, the field that was developed aiming at a fruitful application of logical means to mathematics, by then had already achieved one of its most important results: G ¨odel’s Incompleteness Theorems.
The work of Thoralf Skolem on non-standard models of arithmetic can be situated in this fertile period: whereas Dedekind can be considered as one of the pioneers of mathematical logic for his attempt to pursue a logical account of arithmetic, Skolem puts himself forward as a mature mathematical logic scholar who brought up brilliant results so as to break new ground to this field. In his introductory paper to the collection of Skolem’s texts, Hao Wang puts it as follows:
Skolem has made many fundamental contributions to modern investiga- tions on the foundations of mathematics. [...] [Skolem’s] work dealt with mathematical logic before such fragmentations became prevalent and has quite a different flavor from the more abstract current approaches: rather he partic- ipated in the founding of these subjects by introducing initially several of the basic ideas in their concrete and naked form. [wang70, p.17]
Skolem’s research sets off in a period when logic had progressively acquired maturity and its own tools and methods. To this extent, we can say that Skolem had at his disposal a neat formulation of first-order logic together with the most important results pertaining to it. Before G ¨odel proved the completeness of first-order logic106 in 1930, Skolem had proved an important meta-property of first-order theories, today stated as: “if a set of sentences has a model, then it has a denumerable model.”107
106Recall that here completeness stands for “exhaustiveness”, see footnote 16. 107[boolos07, p. 147]
3.1. SKOLEM AND NON-STANDARD MODELS
The theorem was, put forward for the first time by L ¨owenheim in 1915, Skolem mended one mistake in the proof and generalized it in 1920 by means of the application of the ingenious device now known as Skolem functions108.
Skolem’s approach to non-standard models differs, to some extent, from Dedekind’s: the increasing foundational interest and the fundamental and revolutionary results achieved in logic characterise the status of mathematical logic at Skolem’s time. The issue of “non- standard models of mathematical concepts” was discussed by Skolem in connection with the very same 1920 results concerning models of first-order theories. In fact, Skolem takes note in 1923 that, if there is a model satisfying the axioms of set theory (which are stated in first-order language), then, because of his theorem of 1920 (well-known as the L ¨owenheim-Skolem theorem), a countable structure will be a model of the axioms as well.
According to Skolem, a serious difficulty now lies in the fact that a set-theoretic ax- iomatization is provided in order to capture in a natural way the mathematical structure represented by set theory whose main feature is the “uncountability” of its domain. Thus, by his 1920 result, first order axiomatization fails to render this fundamental property due to the presence of models whose domain is not “uncountable”. In this light, Skolem in- troduces the notion of “non-standard models”109: a model which satisfies the axioms of set theory but does not exhibit the natural “uncountability” that it would be supposed to have.110 This apparent contradiction is known in the literature as Skolem’s paradox111.
From that moment on, Skolem began to question the formal framework as a reli- able foundation for mathematics. For instance, the so-called Skolem paradox is used by Skolem to claim that set-theory cannot serve as a “foundation of mathematics”.
However, this attitude is not motivated by a rejection of foundational research as a pseudo-issue, but rather, by means of his criticisms, Skolem hopes to find the solid and unshakable foundations on which mathematical methods can be build. In this light, he also questions infinitary methodes as not safe enough.112
The 1923 remark on set theory represents the starting point of Skolem’s discussion on non-standard models: once again, the notion ofnon-standardembodies the idea of unde- sirable results suggesting a revision of the assumptions in the light of what we would like to achieve. Nonetheless, Skolem does not restrict himself only to the set-theoretical case, but he envisages that this issue also concerns other mathematical structures. As Jens Erik Fenstad reports in the biographical note “Thoralf Albert Skolem in Memoriam”:
108For a definition of Skolem functions, see [wang70, p.19–22]
109Indeed, 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.
110Today we actually know that there are other types of non-standard models of set theory other than the countable ones.
111See [bays09]
3.1. SKOLEM AND NON-STANDARD MODELS
Toward the end of the 1929 paper113 Skolem expressed some doubts as to the complete axiomatizability of mathematical concepts. His scepticism was based on the set-theoretic relativism which follows from the L ¨owenheim- Skolem theorem. In 1929 he could give only some partial results, but in a pa- per from 1934 (and a previous one from 1933114) ¨Uber die Nicht-characterisierbarkeit der Zahlenreihe115he could prove that there is no finite or countably infinite set of sentences in the language of Peano arithmetic which characterises the natural numbers. [fenstad70, p.14]
Before turning to the arithmetical case, we put forward another point on which Skolem and Dedekind seem to differ significantly. As a consequence of the mature status of the research in mathematical logic, we can credit Skolem as the father of non-standard mod- els of arithmetic in a “modern sense”. This claim is not in contradiction with what we have argued about Dedekind.
Dedekind brings upin nucethat non-standard models are different from the intended model of arithmetic, and that it is not possible to ban them as long as we only consider the theoryDT−. However, Dedekind mentions the construction of NMoA just in passing as he steers toward a categorical axiomatization of natural numbers and, in the second place, makes a controversial claim about the possibility of theorems of arithmetic not be- ing preserved in these non-standard models. On the other hand, Skolem is aware that such theorems do actually hold inproperly constructednon-standard models, and that we cannot distinguish these with respect to first-order properties (i.e. both axioms and theo- rems). This enables Skolem to fully recognise NMoA as a relevant issue to foundational research.
To sum up, Skolem’s study of non-standard models of arithmetic differs from Dedekind’s in two respects:
1. Unlike Dedekind, Skolem introduces non-standard models by considering a formal theory for arithmetic, viz. a first order theory which contains the PA axioms and thus is provided with an induction axiom, strong enough to derive at least most of the ordinary theorems of arithmetic;
2. In virtue of the then-developments in mathematical logic, Skolem was able to ad- dress his discussion on non-standard models as a limitative result to some attempted lines of research and corroborate his position by referring to analogous phenomena in other mathematical branches.
Nevertheless, both Skolem and Dedekind agree on one important point: both con- sider NMoA as a limitative result which comes out as a bound to the descriptive task of
113Ed. note: [skolem29] 114Ed. note: [skolem33] 115Ed. note: [skolem34]