Cuidar/cuidados

called unintended, that undermine our knowledge of the intended or privileged struc- ture.

4.2

The mathematical phase

Both Dedekind’s and Skolem’s accounts of non-standard models were characterised by the following: the attitude to non-standard models of arithmetic as shortcomings of the first-order axiomatization. Since their attempts were philosophically oriented, we called that period the ”philosophical phase” of non-standard models. Although the existence of non-standard models was considered as an embarrassing result from a philosophical point of view, in 1950 Leon Henkin showed his interest toward these models in their own by studying their order type.

Henkin’s paper paved the way for the mathematical results on non-standard models culminating in the 1960s in the first important results : the Tennenbaum and MacDowell- Specker Theorems. With Henkin we witness a radical change with respect to the consid- eration of non-standard models of arithmetic. So we call this period ”the mathematical phase of NMoA”.

4.2.1 From the philosophical phase to the mathematical one

In the previous chapters, we tried to give an account of non-standard models of arith- metic through Dedekind’s and Skolem’s work. Although they did not have a great deal to say about non-standard models themselves, we noted that both Dedekind and Skolem rendered them as an end extension of the standard model.181

The 1950s have witnessed a shift in the way of conceiving non-standard models of arithmetic. Not only does Henkin provide a new terminology as we discussed, but his paper can also be considered as the landmark for the study of non-standard models, as Abraham Robinson and Craig Smorynski put forth forcefully:

Skolem was interested only in showing that no axiomatic system specified in a formal language [...] can characterize the natural numbers categorically; and he did not concern himself further with the properties of the structures whose existence he had established. In due course these and similar struc- tures became known asnonstandard models of arithmetic[...] [robinson67, p.818] Skolem’s goal in constructing nonstandard models was philosophical: He aimed to shew that first-order logic could not characterise the number series; he did not care to start a new subject. Until the 1960s, this was generally

181_{Yet, we have to be careful to use the wordextensionin this context: while Dedekind described non-} standard models as simply supersets, Skolem actually shows that there exists a non-standard model which is a proper extension of the standard one.

4.2. THE MATHEMATICAL PHASE

the case– nonstandard models of arithmetic were either objects of philosoph- ical interest or tools, not objects of mathematical interest in their own right. [smorynski84, p. 3]

While Dedekind and Skolem consider non-standard models as undesired shortcom- ings, Henkin paves the way for acknowledging full dignity and interest to these entities. Once we know that non-standard models exist, it is natural to ask how different they are from the standard one. With Skolem’s result, we can only say that non-standard models of arithmetic are proper extensions of the standard one, as depicted in diagram 4.1182.

Figure 4.1: A non-standard model of arithmetic

Nevertheless, the mathematical phase is indeed characterised by the interest in inves- tigating such non-standard models into detail. In particular, the study of the order type of the models tries to give an answer to the question to which extent NMoA are different from the standard ones.

4.2.2 Henkin and the order type of NMoA

Henkin is interested in providing the ”size” of non-standard models in terms of ”order types”, as Smorynski continues:

The major counterexample to this [i.e. non-standard models as either ob- jects of philosophical interest or tools] was an observation made by Leon

4.2. THE MATHEMATICAL PHASE

Henkin in his paper Henkin 1950 on the Completeness Theorem for Type The- ory. He announced the order type of a nonstandard model of arithmetic to be

ω+ (ω∗+ω)η, whereηis a dense linear order. [smorynski84, p. 3]

Indeed, in this respect Henkin considers only the class of denumerable non-standard models:

It, therefore, becomes of practical interest to number-theorists to study the structure of such models [i.e. non-standard models]. A detailed investigation of these numerical structures is beyond the scope of the present paper. As an example, however, we quote one simple result: Every non-standarddenumer- ablemodel for the Peano axioms has the order typeω+ (ω∗+ω)ηwhereηis the type of the rationals. [henkin50, p. 91]183

Henkin restricts himself to the class of denumerable non-standard models. This is due to the complex picture obtained by a closer scrutiny of the class of non-standard models. In fact, even a quick glance at a non-standard model reveals a very rich structure.

Even though the diversity among non-standard models is so vast that no coherent picture in terms of a relative classification can be painted, we can consider some well defined as well as more tameable subclasses of non-standard models.

Going back to the order type, Henkin claims that any countable non-standard model of arithmetic has order typeω+ (ω∗+ω)·η. Recall thatω is the order type of the stan- dard natural numbers,ω∗is the dual order (an infinite decreasing sequence) andηis the order type of the rational numbers. In diagram 4.2184 we have a picture of how these denumerable non-standard models look.

Figure 4.2: The order typeN+Z·A

To sum up, [henkin50] can be considered as a crucial paper to the radical change of the role of non-standard models. Henkin was indeed interested in studying those models by providing, for example, their ”size” in terms of ”order types”. Thus, he paved the way for the study of non-standard models as a mathematical interest in its own right.

183_{Also Robinson recognises Henkin as the one who provided the order type of the non-standard models.} See [robinson96, p. 88]

184_{The diagram is taken form [kaye91, p. 74]. Note that in the diagramω}∗

+ωis replaced for convenience with the order type of the integers, denoted byZ.