# Get e-book Was sind und was sollen die Zahlen? (German Edition)

An intermediate version of Dedekind's theory was also published separately in a French translation Dedekind Further works by him include: a long article on the theory of algebraic functions, written jointly with Heinrich Weber Dedekind ; and a variety of shorter pieces in algebra, number theory, complex analysis, probability theory, etc. All of these were re-published, together with selections from his Nachlass , in Dedekind — As this brief chronology indicates, Dedekind was a wide-ranging and very creative mathematician, although he tended to publish slowly and carefully.

It also shows that he was part of a distinguished tradition in mathematics, extending from Gauss and Dirichlet through Riemann, Dedekind himself, Weber, and Cantor in the nineteenth century, on to David Hilbert, Ernst Zermelo, Emmy Noether, B. With some partial exceptions, these mathematicians did not publish explicitly philosophical treatises. At the same time, all of them were very sensitive to foundational issues in mathematics understood in a broad sense, including the choice of basic concepts, the kinds of reasoning to be used, and the presuppositions build into them.

Consequently, one can find philosophically pregnant remarks sprinkled through their works, as exemplified by Dedekind and a. Not much is known about other intellectual influences on Dedekind, especially philosophical ones. His short biography of Riemann Dedekind a also contains a reference to the post-Kantian philosopher and educator J. Fichte, in passing Scharlau However, he does not aligns himself explicitly with either of them, nor with any other philosopher or philosophical school.

In fact, little is known about which philosophical texts might have shaped Dedekind's views, especially early on. A rare piece of information we have in this connection is that he became aware of Gottlob Frege's most philosophical work, Die Grundlagen der Arithmetik published in , only after having settled on his own basic ideas; similarly for Bernard Bolzano's Paradoxien des Unendlichen Dedekind a, preface to the second edition. Then again, German intellectual life at the time was saturated with discussions of Kantian and Neo-Kantian views, including debates about the role of intuition for mathematics, and there is evidence that Dedekind was familiar with at least some of them.

But their roots go deeper, all the way down, or back, to the discovery of incommensurable magnitudes in Ancient Greek geometry Jahnke , ch. The Greeks' response to this startling discovery culminated in Eudoxos' theory of ratios and proportionality, presented in Chapter V of Euclid's Elements Mueller , ch.

This theory brought with it a sharp distinction between discrete quantities numbers and continuous quantities magnitudes , thus leading to the traditional view of mathematics as the science of number, on the one hand, and of magnitude, on the other hand. Dedekind's first foundational work concerns, at bottom, the relationship between the two sides of this dichotomy. An important part of the dichotomy, as traditionally understood, was that magnitudes and ratios of them were not thought of as numerical entities, with arithmetic operations defined on them, but in a more concrete geometric way as lengths, areas, volumes, angles, etc.

More particularly, while Eudoxos' theory provides a contextual criterion for the equality of ratios, it does not include a definition of the ratios themselves, so that they are not conceived of as independent objects Stein , Cooke Such features do little harm with respect to empirical applications of the theory; but they lead to inner-mathematical tensions when solutions to various algebraic equations are considered some of which could be represented numerically, others only geometrically. This tension came increasingly to the fore in the mathematics of the early modern period, especially after Descartes' integration of algebra and geometry.

What was called for, then, was a unified treatment of discrete and continuous quantities. More directly, Dedekind's essay was tied to the arithmetization of analysis in the nineteenth century—pursued by Cauchy, Bolzano, Weierstrass, and others—which in turn was a reaction to tensions within the differential and integral calculus, introduced earlier by Newton, Leibniz, and their followers Jahnke , chs.

Yet this again, or even more, led to the need for a systematic characterization of various quantities conceived of as numerical entities, including a unified treatment of rational and irrational numbers. Dedekind faced this need directly, also from a pedagogical perspective, when he started teaching classes on the calculus at Zurich in Dedekind , preface.

Moreover, the goal for him was not just to supply a unified and rigorous account of rational and irrational numbers; he also wanted to do so in a way that established the independence of analysis from mechanics and geometry, indeed from intuitive considerations more generally. This reveals a further philosophical motivation for Dedekind's work on the foundations of analysis, not unconnected with the mathematics involved, and it is natural to see an implicit anti-Kantian thrust in it. Finally, the way in which to achieve all of these objectives was to relate arithmetic and analysis closely to each other, indeed to reduce the latter to the former.

The crucial issue, or the linchpin, for him was the notion of continuity. To get clearer about that notion, he compared the system of rational numbers with the points on a geometric line. Once a point of origin, a unit length, and a direction have been picked for the latter, the two systems can be correlated systematically: each rational number corresponds, in a unique and order-preserving way, with a point on the line. 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? 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.

## Imperative zahlen (pay) | All forms, grammar, examples, voice output

These axioms are thus properly called the Dedekind-Peano axioms. Peano, who published his corresponding work in , acknowledged Dedekind's priority; cf. 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. 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.

We already touched on a specific way in which this comes up in Dedekind's work. Namely, in Was sind und was sollen die Zahlen? As already noted, Dedekind goes beyond considering only sets of numbers in his essay. This is a significant extension of the notion of set, or of its application, but it is not where the main problem lies, as we know now. However, the most problematic feature—and the one Dedekind came to take seriously himself—is a third one: his set theory is subject to the set-theoretic antinomies, including Russell's antinomy.

If any collection of objects counts as a set, then also Russell's collection of all sets that do not contain themselves; but this leads quickly to a contradiction. This news shocked Dedekind initially. Thus he delayed republication of Was sind und was sollen die Zahlen? Russell's antinomy and related problems establish that Dedekind's original conception of set is untenable.

However, they do not invalidate his other contributions to set theory. Dedekind's analysis of continuity, the use of Dedekind cuts in the characterization of the real numbers, the definition of being Dedekind-infinite, the formulation of the Dedekind-Peano axioms, the proof of their categoricity, the analysis of the natural numbers as finite ordinal numbers, the justification of mathematical induction and recursion, and most basically, the insistence on extensional, general notions of set and function, as well as the acceptance of the actual infinite—all of these contributions can be isolated from the set-theoretic antinomies.

As such, they have been built into the very core of contemporary axiomatic set theory, model theory, recursion theory, and other parts of logic.

## Imperative of the verb zahlen

And there are further contributions to set theory we owe to Dedekind. These do not appear in his published writings, but in his correspondence. These letters contain a discussion of Cantor's and Dedekind's respective treatments of the real numbers. But more than that, they amount to a joint exploration of the notions of set and infinity.

This led, at least in part, to Cantor's further study of infinite cardinalities and to his discovery, soon thereafter, that the set of all real numbers is not countable. Aus der Zeitung : Some slightly edited passive examples from a German newspaper with the passive verb bolded. To conjugate the verb forms in the passive voice, you use "werden" in its various tenses. Below are English-German examples of the passive in six different tenses, in the following order: present, simple past Imperfekt , present perfect Perfekt , past perfect, future and future perfect tenses.

The passive voice is used more frequently in written German than in spoken German. German also uses several active-voice substitutes for the passive voice. Below are more examples of passive substitutes in German. Share Flipboard Email. While many of Dedekind's contributions to mathematics and its foundations are thus common knowledge, they are seldom discussed together.

In particular, his foundational writings are often treated separately from his other mathematical ones. This entry provides a broader and more integrative survey. The main focus will be on his foundational writings, but they will be related to his work as a whole. Another goal of the entry is to establish the continuing relevance of his contributions to the philosophy of mathematics, whose full significance has only started to be recognized. This is especially so with respect to methodological and epistemological aspects of Dedekind's approach, which ground the logical and metaphysical views that emerge in his writings.

Richard Dedekind was born in Brunswick Braunschweig , a city in northern Germany, in Much of his education took place in Brunswick as well, where he first attended school and then, for two years, the local technical university. He wrote a dissertation in mathematics under Gauss, finished in As was customary, he also wrote a second dissertation Habilitation , completed in , shortly after that of his colleague and friend Bernhard Riemann.

During that time he was strongly influenced by P. Later, Dedekind did important editorial work for Gauss, Dirichlet, and Riemann. He returned to Brunswick in , where he became professor at the local university and taught until his retirement in In this later period, he published most of his major works.

He also had interactions with other important mathematicians; thus he was in correspondence with Georg Cantor, collaborated with Heinrich Weber, and developed an intellectual rivalry with Leopold Kronecker. He stayed in his hometown until the end of his life, in Landau , ch. Dedekind's main foundational writings are: Stetigkeit und irrationale Zahlen and Was sind und was sollen die Zahlen? Equally important, as emphasized by historians of mathematics, is his work in algebraic number theory. The latter text was based on Dedekind's notes from Dirichlet's lectures, edited further by him, and published in a series of editions.

It is in his supplements to the second edition, from , that Dedekind's famous theory of ideals was first presented. He modified and expanded it several times, with a fourth edition published in Lejeune-Dirichlet , Dedekind An intermediate version of Dedekind's theory was also published separately in a French translation Dedekind Further works by him include: a long article on the theory of algebraic functions, written jointly with Heinrich Weber Dedekind ; and a variety of shorter pieces in algebra, number theory, complex analysis, probability theory, etc.

All of these were re-published, together with selections from his Nachlass , in Dedekind — As this brief chronology indicates, Dedekind was a wide-ranging and very creative mathematician, although he tended to publish slowly and carefully. It also shows that he was part of a distinguished tradition in mathematics, extending from Gauss and Dirichlet through Riemann, Dedekind himself, Weber, and Cantor in the nineteenth century, on to David Hilbert, Ernst Zermelo, Emmy Noether, B.

With some partial exceptions, these mathematicians did not publish explicitly philosophical treatises. At the same time, all of them were very sensitive to foundational issues in mathematics understood in a broad sense, including the choice of basic concepts, the kinds of reasoning to be used, and the presuppositions build into them. Consequently, one can find philosophically pregnant remarks sprinkled through their works, as exemplified by Dedekind and a. Not much is known about other intellectual influences on Dedekind, especially philosophical ones.

His short biography of Riemann Dedekind a also contains a reference to the post-Kantian philosopher and educator J. Fichte, in passing Scharlau However, he does not aligns himself explicitly with either of them, nor with any other philosopher or philosophical school. In fact, little is known about which philosophical texts might have shaped Dedekind's views, especially early on.

A rare piece of information we have in this connection is that he became aware of Gottlob Frege's most philosophical work, Die Grundlagen der Arithmetik published in , only after having settled on his own basic ideas; similarly for Bernard Bolzano's Paradoxien des Unendlichen Dedekind a, preface to the second edition. Then again, German intellectual life at the time was saturated with discussions of Kantian and Neo-Kantian views, including debates about the role of intuition for mathematics, and there is evidence that Dedekind was familiar with at least some of them.

But their roots go deeper, all the way down, or back, to the discovery of incommensurable magnitudes in Ancient Greek geometry Jahnke , ch. The Greeks' response to this startling discovery culminated in Eudoxos' theory of ratios and proportionality, presented in Chapter V of Euclid's Elements Mueller , ch. This theory brought with it a sharp distinction between discrete quantities numbers and continuous quantities magnitudes , thus leading to the traditional view of mathematics as the science of number, on the one hand, and of magnitude, on the other hand.

Dedekind's first foundational work concerns, at bottom, the relationship between the two sides of this dichotomy. An important part of the dichotomy, as traditionally understood, was that magnitudes and ratios of them were not thought of as numerical entities, with arithmetic operations defined on them, but in a more concrete geometric way as lengths, areas, volumes, angles, etc.

More particularly, while Eudoxos' theory provides a contextual criterion for the equality of ratios, it does not include a definition of the ratios themselves, so that they are not conceived of as independent objects Stein , Cooke Such features do little harm with respect to empirical applications of the theory; but they lead to inner-mathematical tensions when solutions to various algebraic equations are considered some of which could be represented numerically, others only geometrically. This tension came increasingly to the fore in the mathematics of the early modern period, especially after Descartes' integration of algebra and geometry.

What was called for, then, was a unified treatment of discrete and continuous quantities. More directly, Dedekind's essay was tied to the arithmetization of analysis in the nineteenth century—pursued by Cauchy, Bolzano, Weierstrass, and others—which in turn was a reaction to tensions within the differential and integral calculus, introduced earlier by Newton, Leibniz, and their followers Jahnke , chs. Yet this again, or even more, led to the need for a systematic characterization of various quantities conceived of as numerical entities, including a unified treatment of rational and irrational numbers.

Dedekind faced this need directly, also from a pedagogical perspective, when he started teaching classes on the calculus at Zurich in Dedekind , preface. Moreover, the goal for him was not just to supply a unified and rigorous account of rational and irrational numbers; he also wanted to do so in a way that established the independence of analysis from mechanics and geometry, indeed from intuitive considerations more generally. This reveals a further philosophical motivation for Dedekind's work on the foundations of analysis, not unconnected with the mathematics involved, and it is natural to see an implicit anti-Kantian thrust in it.

Finally, the way in which to achieve all of these objectives was to relate arithmetic and analysis closely to each other, indeed to reduce the latter to the former. The crucial issue, or the linchpin, for him was the notion of continuity. To get clearer about that notion, he compared the system of rational numbers with the points on a geometric line. Once a point of origin, a unit length, and a direction have been picked for the latter, the two systems can be correlated systematically: each rational number corresponds, in a unique and order-preserving way, with a point on the line.

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.

### References

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.

- Stochastic Relations: Foundations for Markov Transition Systems (Chapman & Hall/CRC Studies in Informatics Series).
- "zahlen" English translation;
- Southern Cultures, 16:1.

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? 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.

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.

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.

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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.