Abstract
This chapter discusses the contributions of Richard Dedekind and Giuseppe Peano to the foundations of arithmetic. Although Dedekind presented basic results very clearly and succinctly, he lacked terminology and notation for the membership relation. Was sind und was sollen die Zahlen (WSZ) was the first work to thematise the concept of mapping and to discuss its basic theory. The most original and profound theoretical development in WSZ is the so-called theory of chains, elaborated in a section on internal mappings. The concept of chain of a subset was obtained by analyzing and generalizing the conditions that an internal mapping must satisfy for making proofs by induction possible. One of the distinguishing characteristics of WSZ is that Dedekind proceeded to investigate the required concepts in great generality. While even today some authors regard Dedekind's approach as formal and too abstract, others welcome it as a key instance of structural reasoning in mathematics. Generally speaking, the axiomatic treatment of the (finite) ordinals and of transfinite induction is closely related to the work of Dedekind.
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