But a further question then arises: Does each point on the line correspond to a rational number? Namely, if we divide the whole system of rational numbers into two disjoint parts while preserving their order, is each such division determined by a rational number? The answer is no, since some correspond to irrational numbers e. In this explicit, precise sense, the system of rational numbers is not continuous, i. For our purposes several aspects of Dedekind's procedure, at the start and in subsequent steps, are important cf.
As indicated, Dedekind starts by considering the system of rational numbers seen as a whole.
In his next step—and proceeding further along set-theoretic and structuralist lines—Dedekind introduces the set of arbitrary cuts on his initial system, thus working essentially with the bigger and more complex infinity of all subsets of the rational numbers the full power set. It is not the cuts themselves with which Dedekind wants to work in the end, however. Dedekind b, b. Those objects, together with an order relation and arithmetic operations defined on them in terms of the corresponding cuts , form the crucial system for him.
Next, two properties of the new system are established: The rational numbers can be embedded into it, in a way that respects the order and the arithmetic operations a corresponding field homomorphism exists ; and the new system is continuous, or line-complete, with respect to its order. What we get, overall, is the long missing unified criterion of identity for rational and irrational numbers, both of which are now treated as elements in an encompassing number system isomorphic to, but distinct from, the system of cuts. Finally Dedekind indicates how explicit and straightforward proofs of various facts about the real numbers can be given along such lines, including ones that had been accepted without rigorous proof so far.
These include: basic rules of operation with square roots; and the theorem that every increasing bounded sequence of real numbers has a limit value a result equivalent, among others, to the more well-known intermediate value theorem.
Dedekind's published this account of the real numbers only in , fourteen years after developing the basic ideas on which it relies. Most familiar among their alternative treatments is probably Cantor's, also published in The system of such classes of sequences can also be shown to have the desired properties, including continuity. Like Dedekind, Cantor starts with the infinite set of rational numbers; and Cantor's construction again relies essentially on the full power set of the rational numbers, here in the form of arbitrary Cauchy sequences.
In such set-theoretic respects the two treatments are thus equivalent. What sets apart Dedekind's treatment of the real numbers, from Cantor's and all the others, is the clarity he achieves with respect to the central notion of continuity. His treatment is also more maturely and elegantly structuralist, in a sense to be spelled out further below. Providing an explicit, precise, and systematic definition of the real numbers constitutes a major step towards completing the arithmetization of analysis.
Further reflection on Dedekind's procedure and similar ones leads to a new question, however: What exactly is involved in it if it is thought through fully, i. As noted, Dedekind starts with the system of rational numbers; then he uses a set-theoretic procedure to construct, in a central step, the new system of cuts out of them. This suggests two sub-questions: First, how exactly are we to think about the rational numbers in this connection? Second, can anything further be said about the relevant set-theoretic procedures and the assumptions behind them?
In his published writings, Dedekind does not provide an explicit answer to our first sub-question. What suggests itself from a contemporary point of view is that he relied on the idea that the rational numbers can be dealt with in terms of the natural numbers together with some set-theoretic techniques. It seems that these constructions were familiar enough at the time for Dedekind not to feel the need to publish his sketches.
There is also a direct parallel to the construction of the complex numbers as pairs of real numbers, known to Dedekind from W. Hamilton's works, and more indirectly, to the use of residue classes in developing modular arithmetic, including in Dedekind For the former cf. This leads to the following situation: All the material needed for analysis, including both the rational and irrational numbers, can be constructed out of the natural numbers by set-theoretic means. But then, do we have to take the natural numbers themselves as given; or can anything further be said about those numbers, perhaps by reducing them to something even more fundamental?
Many mathematicians in the nineteenth century were willing to assume the former. This is the main goal of Was sind und was sollen die Zahlen? The Nature and Meaning of Numbers , or more literally, What are the numbers and what are they for? Another goal is to answer the second sub-question left open above: whether more can be said about the set-theoretic procedures used.
But what are the basic notions of logic?
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These notions are, indeed, fundamental for human thought—they are applicable in all domains, indispensable in exact reasoning, and not reducible further. While thus not definable in terms of anything even more basic, the fundamental logical notions are nevertheless capable of being elucidated, thus of being understood better. Part of their elucidation consists in observing what can be done with them, including how arithmetic can be reconstructed in terms of them more on other parts below.
For Dedekind, that reconstruction starts with the consideration of infinite sets, as in the case of the real numbers, but now in a generalized and more systematic manner. Dedekind does not just assume, or simply postulate, the existence of infinite sets; he tries to prove it. He also does not just presuppose the concept of infinity; he defines it in terms of his three basic notions of logic, as well as the definable notions of subset, union, intersection, etc.
A set can then be defined to be finite if it is not infinite in this sense. What it means to be simply infinite can now be captured in four conditions: Consider a set S and a subset N of S possibly equal to S. While at first unfamiliar, it is not hard to see that these Dedekindian conditions are a notational variant of Peano's axioms for the natural numbers.
In particular, condition ii is a version of the axiom of mathematical induction. These axioms are thus properly called the Dedekind-Peano axioms. Peano, who published his corresponding work in , acknowledged Dedekind's priority; cf.
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Given these preparations, the introduction of the natural numbers can proceed as follows: First, Dedekind proves that every infinite set contains a simply infinite subset. Then he establishes that any two simply infinite systems, or any two models of the Dedekind-Peano axioms, are isomorphic so that the axiom system is categorical.
Third, he notes that, as a consequence, exactly the same arithmetic truths hold for all simple infinities; or closer to Dedekind's actual way of stating this point, any truth about one of them can be translated, via the isomorphism, into a corresponding truth about the other.
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In those respects, each simply infinity is as good as any other. As we saw, this last step has an exact parallel in the case of the real numbers see again Dedekind b. However, in the present case Dedekind is more explicit about some crucial aspects. In particular, the identity of the newly created objects is determined completely by all arithmetic truths, i.
A set turns out to be finite in the sense defined above if and only if there exists such an initial segment of the natural numbers series. Dedekind rounds off his essay by showing how several basic, and formerly unproven, arithmetic facts can now be proved too. As indicated, set-theoretic assumptions and procedures already inform Dedekind's Stetigkeit und irrationale Zahlen. In particular, the system of rational numbers is assumed to be composed of an infinite set; the collection of arbitrary cuts of rational numbers is treated as another infinite set; and when supplied with an order relation and arithmetic operations on its elements, the latter gives rise to a new number system.
Parallel moves can be found in the sketches, from Dedekind's Nachlass , of how to introduce the integers and the rational numbers. Once more we start with an infinite system, here that of all the natural numbers, and new number systems are constructed out of it set-theoretically although the full power set is not needed in those cases.
Finally, Dedekind uses similar set-theoretic techniques in his other mathematical work as well e. It should be emphasized that the application of such techniques was quite novel and bold at the time. While a few mathematicians, such as Cantor, used them too, many others, like Kronecker, rejected them. What happens in Was sind und was sollen die Zahlen?
Dedekind not only presents set-theoretic definitions of various mathematical notions, he also adds a systematic reflection on the means used thereby and he expands that use in certain respects. Consequently, the essay constitutes an important step in the rise of modern set theory.
We already saw that Dedekind presents the notion of set, together with those of object and function, as fundamental for human thought. Here an object is anything for which it is determinate how to reason about it, including having definite criteria of identity Tait Sets are a kind of objects about which we reason by considering their elements, and this is all that matters about them.
In other words, sets are to be identified extensionally, as Dedekind is one of the first to emphasize. Even as important a contributor to set theory as Bertrand Russell struggles with this point well into the twentieth century. Dedekind is also among the first to consider, not just sets of numbers, but sets of various other objects as well.
Functions are to be conceived of extensionally too, as ways of correlating the elements of sets. Unlike in contemporary set theory, however, Dedekind does not reduce functions to sets. Not unreasonably, he takes the ability to map one thing onto another, or to represent one by the other, to be fundamental for human thought; see Dedekind a, preface.
Another important aspect of Dedekind's views about functions is that, with respect to their intended range, he allows for arbitrary functional correlations between sets of numbers, indeed between sets of objects more generally. He thus rejects previous, often implicit restrictions of the notion of function to, e. That is to say, he works with a generalized notion of function. Dedekind's notion of set is general in the same sense. Such general notions of set and function, together with the acceptance of the actual infinite that gives them bite, were soon attacked by finitistically and constructively oriented mathematicians like Kronecker.
Dedekind defended his approach by pointing to its fruitfulness Dedekind a, first footnote, cf. But eventually he came to see one feature of it as problematic: his implicit acceptance of a general comprehension principle another sense in which his notion of set is unrestricted.