Does non-standard amount to unintended?

As discussed above, the standard/non-standard distinction came to be applied both to models and to the interpretation of second-order quantifiers. Still, we have not men- tioned which features lie behind a standard and non-standard model or interpretation.

4.1. AN EXEGETICAL AND HISTORICAL CLARIFICATION

In other words, we want to elaborate more on the reason that would lead us to call a model or an interpretation ”non-standard” rather than ”standard”.

For instance, the standard interpretation of second-order quantifiers does not seem to be historically motivated, as Hintikka suggests

A terminological point is in order here. By calling one of the two con- trasting interpretations ”standard”, we are not passing any judgement as to which view ought to be adopted or which one is historically the usual one. [hintikka92, p. 147, my emphasis]

Hintikka points out that the standard semantics of second-order logic relies on the notion of arbitrary set and arbitrary function that in the past was far from being the standard approach in mathematics.

The idea of standard interpretation is virtually identical with the idea of a completely arbitrary function. Hence the gradual development of what in effect was the notion of standard interpretation can be partially followed by tracing the history of the notion of a (completely) arbitrary function. [hintikka95, p. 110]

Conversely, as far as standard and non-standard models are concerned, Hintikka does not go further than saying that with respect to arithmetic the ”standard model” is nothing but the intended model:

[...] it is far from clear how Henkin’s notion of standard interpretation or standard model is related to logician’s idea of the standard model of such first- order theories as elementary arithmetic, in which usage ”standard model” means simply ”intended model”. [hintikka95, p. 106]

That, Along Hintikka’s lines, NMoA amount to the unintended models of arithmetic follows at once. Although in the previous chapters we agreed with Hintikka and we adopted interchangeably intended and standard as well as unintended and non-standard, here we want to say something more about the interplay between these two dichotomies. In this respect, Wang has once again a leading role in the way non-standard models came to be developed in philosophy. In his 1957 paper, Wang drew for the first time the important connection between non-standard and unintended models.179

To some extent, Wang puts himself as a spokesperson of thephilosophical phaseof non- standard models as we described previously: he presents the existence of non-standard models as a consequence of the failed attempt to capture completely the intended model.

4.1. AN EXEGETICAL AND HISTORICAL CLARIFICATION

Along these lines, to talk about intended and unintended models is natural insofar as we aim at describing a certain mathematical structure. To put this in another way, when a theory is formulated in order to capture such a structure, the model (or mod- els) exemplifying this structure is called the intended or standard model (or intended or standard models). The others are called unintended or non-standard models. The de- scriptive incompleteness of a theory amounts to the fact that the theory does not rule out all non-standard models.

However, if the descriptive task of logic is put aside, the expressions intended and unintended will come out as vacuous.

In fact, as soon as we decide on the structure to describe, we are able to speak of the intended model (viz. the model that we stipulated as the one our study is all about), and the unintended models (viz. models that represent the outcome of a description ”not yet complete”). The basis of the intended-unintended terminology is the ability to single out a model as privileged with respect to the others. Outside the descriptive task of logic, we have no need in picking out a certain model rather than another.

On the other hand, the ”standard / non-standard” terminology can be seen as math- ematically nuanced. Along these lines, non-standard models represent entities indepen- dently of the descriptive task, so that we can decide to study non-standard models as models in themselves. In accordance with today’s mathematical practice, we say that the standard model of arithmetic has order typeωwhile a denumerable non-standard model has a completely different order type than the standard one.180 In this case we have no privileged model, but we catalogue and name the different classes of models in virtue of their features.

In the previous chapters we have mainly dealt with an exclusively descriptive frame- work. Thus, we assumed for convenience that non-standard (or standard) models amount to unintended (or intended) ones.

Here, according to what we have just suggested, we can conclude that the two di- chotomies can also be viewed as pertaining to two distinct dimensions: one ontological and one epistemic. On the one hand, the terms ”standard” and ”non-standard” can be taken to range over classes of arithmetical models as considered in virtue of their prop- erties. In this sense this distinction is viewed as ontological since we denote the models per sein order to study how they actually are. In this case the name ”standard” is due to historical reasons, namely the class that came to be studied prior to the non-standard ones.

On the other hand, the ”intended-unintended” distinction is epistemic as long as we consider the way we come to know the arithmetical models. In other words, we assume that we have access to a privileged class of models, called intended, and insofar as we describe it as completely as possible, we may come to know some other kind of models,