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Dedekind’s theory of numbers

Dedekind’s set theory is equivalent to Peirce’s and, consequently, so is his theory of numbers. This equivalence of Peirce’s theory of natural numbers to that of Dedekind (as well as that of Giuseppe Peano’s 1889 Arithmetices principia [60]) was demonstrated by Shields (in [93] and [94]).

There are only two significant differences between the development of number theory by Dedekind and by Pierce:

• Dedekind started from infinite sets rather than finite sets in defining natural numbers
• Dedekind is explicitly and specifically concerned with the real number continuum, that is, with infinite sets.

Frege and Dedekind were focused on “reducing” the natural numbers and arithmetic to “logic”. This is the main goal of Dedekind's Was sind und was sollen die Zahlen?

The technical centerpiece of Dedekind’s mathematical work was in number theory, especially algebraic number theory. His primary motivation was to provide a foundation for mathematics and in particular to find a rigorous definition of real numbers and of the real-number continuum upon which to establish mathematical analysis in the style of Karl Weierstrass. This means that he sought to axiomatize the theory of numbers based upon that rigorous definition of the real numbers and the construction of the real number system and the continuum which could be employed in defining the theory of limits of a function for use in the differential and integral calculus, real analysis, and related areas of function theory. His concern, in short, was with the rigorization and arithmetization of analysis.

One goal of Dedekind’s of Was sind und was sollen die Zahlen? was to answer the question, What more can be said about the set-theoretic procedures used? For Dedekind, again like for Frege, these procedures are founded in “logic”. But then, what are the basic notions of logic?

• ‘’Ordinal numbers’’ are used to count elements and place them in a succession; in such cases they correspond to Enligsh expressions such as first, second, third... and so forth.
• ‘’Cardinal numbers’’ are used, loosely speaking, to count how many elements of some kind there are: one cat, two dogs, three horses, and so forth.

For Dedekind, numbers are essentially ordinals.

Dedekind’s efforts in foundations did not stop with the reduction of all mathematics to arithmetic, thinking that both the concepts and the rules of arithmetic itself needed clarification -- principally through logic and set theory.

The ultimate basis of a mathematician’s knowledge is, according to Dedekind, the clarification of the concept of natural numbers (viz., positive integers) in a non-mathematical fashion, which involves this twofold task:

1. to define numerical concepts (natural numbers) through logical ones
2. to characterize mathematical induction (the passage from n to n+1) as a logical inference.

Dedekind’s methods led him to develop a “set-theoretic” style of axiomatic analysis that is quite different from the work of Peano on the natural numbers, or that of Hilbert on geometry and the real numbers…. To clarify the matter, let me remind you of Dedekind’s axioms:

A simply infinite set $N$ has a distinguished element $e$ and an ordering mapping $ϕ$ such that

1. $ϕ(N) ⊆ N$
2. $e \notin ϕ(N)$
3. $N = ϕ0(e)$, i.e. $N$ is the $ϕ-chain of the unitary set $\{e\}$ ::# $ϕ$ is an injective mapping from $N$ to $N$, i.e. if $ϕ(a) = ϕ(b)$ then $a = b$. Leaving aside axioms 2. and 4., which are more easy to assimilate to Peano’s axioms, the other two axioms are characteristically set-theoretic in the intended sense, and not elementary as most of Peano’s and Hilbert’s axioms. Peano tended to impose conditions on the behaviour of his individuals, the natural numbers, and the operations on them. These are elementary conditions which most often are amenable to formalization within first-order logic. Dedekind establishes structural conditions on subsets of the (structured) sets he is defining, on the behaviour of relevant maps, or both things at a time. Axiom 1. says that $N$ is closed under the map $ϕ$, Axiom 3. says that $N$ is the minimal closure of the unitary set $\{e\}$ under $ϕ$. Such axioms are non-elementary and tend to require second-order logic for their formalisation'"`UNIQ--ref-00000007-QINU`"' A set of objects is infinite—“Dedekind-infinite”, as we now say—if it can be mapped one-to-one onto a proper subset of itself.'"`UNIQ--ref-00000008-QINU`"' “What are the mutually independent fundamental properties of the sequence N, that is, those properties that are not derivable from one another but from which all others follow? And how should we divest these properties of their specifically arithmetic character so that they are subsumed under more general notions and under activities of the understanding without which no thinking is possible at all, but with which a foundation is provided for the reliability and completeness of proofs and for the construction of consistent notions and definitions.”'"`UNIQ--ref-00000009-QINU`"' Dedekind's intention was not to axiomatize arithmetic, but to give an "algebraic" characterization of natural numbers as a mathematical structure.'"`UNIQ--ref-0000000A-QINU`"' What it means to be ‘’simply infinite’’ can be captured in four conditions:'"`UNIQ--ref-0000000B-QINU`"' :Consider a set $S$ and a subset $N$ of $S$ (possibly equal to $S$). Then $N$ is called simply infinite if there exists a function $f$ on $S$ and an element $1$ of $N$ such that :# $f$ maps $N$ into itself :# $N$ is the chain of $\{1\}$ in $S$ under $f$ :# $1$ is not in the image of $N$ under $f$ :# $f$ is one-to-one These Dedekindian conditions are a notational variant of the Peano axioms for the natural numbers. In particular, condition 2 is a version of the axiom of mathematical induction. These axioms are thus properly called the Dedekind-Peano axioms.'"`UNIQ--ref-0000000C-QINU`"' As is also readily apparent, any simple infinity will consist of a first element $1$, a second element $f(1)$, a third $f(f(1))$, then $f(f(f(1)))$, and so on, just like any model of the Dedekind-Peano axioms.

Dedekind introduces the natural numbers as follows:

1. he proves that every infinite set contains a simply infinite subset
2. he shows, in contemporary terminology, that any two simply infinite systems, or any two models of the Dedekind-Peano axioms, are isomorphic (so that the axiom system is categorical)
3. he notes that consequently … any truth about one of them can be translated, via the isomorphism, into a corresponding truth about the other.

(As one may put it, all models of the Dedekind-Peano axioms are “logically equivalent”, which means that the axiom system is “semantically complete”; compare Awodey & Reck 2002 and Reck 2003a)

What Dedekind has introduced, along such lines, is the natural numbers conceived of as finite “ordinal” numbers (or counting numbers: the first, the second, etc.). Later he adds an explanation of how the usual employment of the natural numbers as finite “cardinal” numbers (answering to the question: how many?) can be recovered. This is done by using initial segments of the number series as tallies: for any set we can ask which such segment, if any, can be mapped one-to-one onto it, thus measuring its “cardinality”. (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.)