Corrección de algoritmos y vericación de programas

2.6

Dedekind: completeness and categoricity

Having settled the formal notion and the logical framework, we are now able to shed some light on an aspect that we have not made explicit yet, viz. in which sense we should understand Dedekind’s attempt to pursue a “complete characterisation of the nature of the number-sequence”, as he refers in the letter.88

In this section we argue that Dedekind’s notion of “completeness” can be unraveled by means of the considerations that we have amassed so far. In doing so, we show that the meta-theoretical properties, such as categoricity and semantic completeness that we have just defined, have actually to do with Dedekind’s idea of “complete characterisation”.

2.6.1 Complete characterisation

In which way should we understand the word “complete” to which Dedekind refers before presenting the sixth tenet of his letter?

First of all, we must note it would certainly be anachronistic to say that Dedekind exploited notions such as deductive and semantic consequence relations, which were not clearly formulated until G ¨odel’s (some ideas were already in Frege’s texts and ac- cordingly in Russell-Whitehead’s) and Tarski’s works, respectively. Thus, for the lack of “deducibility relation” in his thought, it is reasonable to dismiss that with “complete characterisation” Dedekind means completeness in the sense of “exhaustiveness” in the- orem 2.5.0.2, and therefore any hint at the deductive use of logic in his work.

Still, in spite of the lack of a modern framework and a clear notion of “deducibility”, we advocate that interesting logical aspects can be brought up in Dedekind’s work in connection with the semantic relation.

On page 7, before turning to Dedekind, we put forward the definition of categoricity as the main purpose of the descriptive task of logic in mathematics.

This way of looking at Dedekind’s completeness, viz. as an attempt to use logic in a descriptive way, seems the most plausible. In fact, as we noted so far, Dedekind is extremely clear in which case a characterisation isin-complete, i.e. we acknowledge the existence of a non-standard model of arithmetic. Thus, he seems to suggest that the list of conditions (axioms) is notcompleteinsofar as we are able to construct a model which embodies all the properties specified in such a list, but at the same time such model turns out as admittedly different from the intended structure we were supposed to completely characterise.

In other words, the series of natural numbers is completely characterised (in Dedekind sense) if the axioms we fix on are in such a way that unintended models do not appear. This is the reason why Dedekind comes up with the notion of chain: to rule out non- standard models and thus achieve in characterising completely the series of natural num-

2.6. DEDEKIND: COMPLETENESS AND CATEGORICITY

bers. Along these lines, Dedekind hints at an axiomatization that iscategorical: the set of axioms come to be satisfied by a unique model (the intended one) up to isomorphism.

This is actually the case for the four axiomsα-δ, viz. it is proved to be categorical in NMN by two important theorems.

2.6.2 Complete characterisation as Categoricity

To prove that an axiomatization is categorical, Dedekind has to show that there is indeed an isomorphism between all the models that satisfyα-δ.

In section 2.2.1, we considered Dedekind’s similar transformations and we noted that can be rephrased in modern terminology as injective functions. However, by using simi- lar transformation Dedekind seems to run into one hazard, as Reck stresses

While Dedekind is not completely precise, or at least not completely ex- plicit, about the notion of isomorphism involved here - he defines “similarity” in terms of the existence of a 1-1 (“similar”) function, while what he really needs is a bijective function (1-1 and “onto”) - his proofs of these two theo- rems show that he essentially understands the notion of a simple infinity, thus the Dedekind-Peano Axioms, to be categorical. [reck03, p. 377]

If Dedekind only employs injective (similar) transformation, what he obtains is a mere embedding rather than isomorphism between structures. For this reason, it is crucial that the function has to be not only an injection but also onto, and thus bijective.

However, in contrast to what Reck claims, Dedekind is aware of the necessity of bi- jective rather than injective functions. The controversy lies in the way Dedekind defines “similar transformations” and “similar system”. Whereas asimilar transformationrepre- sents an injective function, two systems are defined to besimilaras follows:

32. Definition. The systemsR, S are said to be similar when there exists such a similar transformationϕofSthatϕ(S) =R, and thereforeϕ(R) =S. We would expect Dedekind to define a similar system as a system provided with a similar function. But Dedekind actually considers a similar function in order to compare a pair of systems. Indeed, the definition not also says that a similar transformation holds between S and R, but that the inverse transformation covers the whole system R. This is to say that,ϕis a function from S to R, such thatf is injective and f[S] = R. In other words,f is taken as a bijection.

This position is corroborated by the way Dedekind defines the inverse transformation introduced in def.26:

aninversetransformation of the system S’, to be denoted byϕ(S0), which consists in this that to every element s’ of S’ there corresponds the transform

2.6. DEDEKIND: COMPLETENESS AND CATEGORICITY

Since s’ and S’ denote the range of the injective function ϕ, it does make sense to consider the inverse functionϕ−1for the elements that fall under the range. Normally, it is not known whether the range ofϕcovers the whole codomain ofϕ. If this is the case the function is indeed bijective, otherwise only injective.

Mainly, Dedekind studies similar transformations and transformations on the system itself. This is the reason why he adopts the convenient way to denote with S’ or s’ the range of S or s under a transformation. For example the notion of chain: the idea is to consider those sets whose similar transformations have the set itself as a range. Still, in the definition of “similar set” what it at issue is a transformation between R and S which is not only injective, but also surjective: “ϕofS thatϕ(S) =R, and thereforeϕ(R) =S”. To sum up, in contrast to Reck’s and other scholars’ idea, the definition of similar sys- tem provided by Dedekind does not fall short as we try to characterise an isomorphism between simply infinite systems.

Once we have set matters straight about that, we can eventually turn to theorems 132 and 133 of NMN which the categoricity of the axioms rests on:

132. Theorem. All simply infinite systems are similar to the number-series N and consequently by (33) also to one another.

133. Theorem. Every system which is similar to a simply infinite system and therefore by (132), (33) to the number-series N is simply infinite.

Theorem 132 claims that any simply infinite system (i.e. a structure that makes true conditionsα-δ) is isomorphic to the model of natural numbers. In short, for Dedekind the difference between them lies in the fact that in the number-series case “we entirely ne- glect the special character of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformationϕ”89_{. Although such a claim would require further discus-}

sion, for our purposes it suffices to say that, arguably, Dedekind urges for a separation between the standard model, the series of the natural numbers, i.e. what we normally mean with positive integers, and all other models that indeed satisfy the same proper- ties. Moreover, by transitivity of similarity90, we also get that two simply infinite systems isomorphic to N are indeed isomorphic to each other.

Theorem 133 completes the argument since it claims that every system which is iso- morphic to a simply infinite system (and therefore to the number series N by thm. 132) comes out as simply infinite as well. Both results entitle Dedekind to conclude as follows:

134. Remark. By the two preceding theorems (132), (133) all simply infinite systems form a class in the sense of (34).

89_{[dedekind01, p. 34]}

90_{”33. Theorem. If R, S are similar systems, then every system Q similar to R is also similar to S.”} [dedekind01, p. 26]

2.6. DEDEKIND: COMPLETENESS AND CATEGORICITY

Figure 2.5: Theorem 132

Thus, we obtain a class91 in the modern sense of the word which includes all the models isomorphic to the standard model N. Therefore, the theory defined by the four conditionsα-δcomes out as categorical since all its models are isomorphic. We call this theory “Dedekind Theory”, DT for short.

But Dedekind continues:

At the same time, with reference to (71), (73) it is clear that every theo- rem regarding numbers, i.e. regarding the elements n of the simply infinite system N set in order by the transformationϕ, and indeed every theorem in which we [...] discuss only such notions as arise from the arrangementϕ,pos- sesses perfectly general validity for every other simply infinite system[...] [dedekind01, p. 48]

Thus, even if without a formal proof as in the previous theorems, Dedekind puts forward the fact that from categoricity of the his logical account of the series of natural numbers it follows that all the theorems (viz. L-formulas) regarding numbers (viz. true in the standard model of arithmetic), once they are formulated in what we would now call a certain formal language,92are valid (viz. with respect to the semantic relation) in every other simply infinite system (viz. a model that satisfies DT). But this states noth- ing but that thesemantically completenessof DT that Dedekind actually derives from the categoricityof DT.

91_{”34. Definition. We can therefore separate all systems intoclassesby putting into a determinate class all} systems Q, R, S, . . . , and only those, that are similar to a determinate system R, therepresentativeof the class; according to (33) the class is not changed by taking as representative any other system belonging to it.” [dedekind01, p. 26]

92_{Roughly speaking, Dedekind hints at this step as long as he talks about a “translation” from the language} of the number-series N to the language of any simply infinite system

2.6. DEDEKIND: COMPLETENESS AND CATEGORICITY

Figure 2.6: Theorem 133-134

Going back to the issue in the remark 2.5.0.10, Dedekind has also something to say about the converse of this implication. Does, for DT, semantically completeness imply categoricity?

Dedekind does not say much about that with respect toDT, but we are likely to say that Dedekind might have answered positively as regards the theoryDT− rather than

DT.

In fact, Dedekind holds a stronger claim, viz. the theoryDT−is both non-categorical and semantically incomplete.93

Consider the theoryDT−. Now, Dedekind proves that the theory DT−, is satisfied by models which are not isomorphic to the standard one, say N, and thereforeDT− is not categorical. In addition, Dedekind claims that we could choose one of these non- standard models such that “scarcely a single theorem of arithmetic would be preserved in it”94. Let M be one of these models. Now, N and M are both models ofDT−but there is some arithmetical theorem which is true in N but not in M. That is, if Nϕthen M2ϕ for almost all arithmetical theoremsϕ. Hence,DT−is semantically incomplete.

93_{From this easily follows that ifDT}−

is non-categorical then it is semantically incomplete. Hence, by contraposition the claim follows.