Title:
Integer construction by induction
Authors:
Krone, Joan; Fressola, Anthony R.
Citation:
Fressola, R.A. and Krone, J. (2000). Integer construction by induction. Conference 00.
Publisher:
Conference 00
DATE ISSUED:
2000
PERMANENT LINK:
http://hdl.handle.net/2374.DEN/5016; http://hdl.handle.net/2374
Type:
Article
Language:
en_US
Description:
In 1889, Giuseppe Peano inductively defined the natural numbers by using the empty set along with a successor function. The natural numbers can be defined inductively primarily because they are well-ordered, a property which is equivalent to that of induction [1]. Inductive systems are especially useful in the area of computing for both reasoning about and implementing algorithms. Moreover, induction lends itself well to certain aspects of automated proving. The natural numbers are just one example of inductive systems. Other inductive systems include string induction, tree induction, and transfinite induction [2, 3]. Although it would seem reasonable to describe sets containing the natural numbers inductively, such as the integers and rational numbers, traditional approaches have not done so. These systems have traditionally been defined as equivalence classes of natural numbers [4]. One reason may be that the integers and rationals are not well-ordered under the usual ordering. This leaves us with an intriguing question: Can the integers and rationals be described inductively? Here we present one possible well-ordering that allows us to define the integers inductively. By introducing several definitions, we are able to prove the common additive properties of the integers, including the associativity and commutativity of addition. This work motivates the future investigation of other systems, such as the rational numbers.
Appears in Collections:
Faculty Publications

Full metadata record

DC FieldValue Language
dc.contributor.authorKrone, Joanen
dc.contributor.authorFressola, Anthony R.en
dc.date.accessioned2013-01-02T17:02:24Zen
dc.date.accessioned2013-12-18T21:05:59Z-
dc.date.available2013-01-02T17:02:24Zen
dc.date.available2013-12-18T21:05:59Z-
dc.date.created2000en
dc.date.issued2000en
dc.identifier.citationFressola, R.A. and Krone, J. (2000). Integer construction by induction. Conference 00.en_US
dc.identifier.urihttp://hdl.handle.net/2374.DEN/5016en
dc.identifier.urihttp://hdl.handle.net/2374-
dc.descriptionIn 1889, Giuseppe Peano inductively defined the natural numbers by using the empty set along with a successor function. The natural numbers can be defined inductively primarily because they are well-ordered, a property which is equivalent to that of induction [1]. Inductive systems are especially useful in the area of computing for both reasoning about and implementing algorithms. Moreover, induction lends itself well to certain aspects of automated proving. The natural numbers are just one example of inductive systems. Other inductive systems include string induction, tree induction, and transfinite induction [2, 3]. Although it would seem reasonable to describe sets containing the natural numbers inductively, such as the integers and rational numbers, traditional approaches have not done so. These systems have traditionally been defined as equivalence classes of natural numbers [4]. One reason may be that the integers and rationals are not well-ordered under the usual ordering. This leaves us with an intriguing question: Can the integers and rationals be described inductively? Here we present one possible well-ordering that allows us to define the integers inductively. By introducing several definitions, we are able to prove the common additive properties of the integers, including the associativity and commutativity of addition. This work motivates the future investigation of other systems, such as the rational numbers.en_US
dc.language.isoen_USen_US
dc.publisherConference 00en_US
dc.relation.ispartofFaculty Publicationsen_US
dc.titleInteger construction by inductionen_US
dc.typeArticleen_US
dc.contributor.institutionDenison Universityen_US
dc.date.digitized2013-01-02en
dc.contributor.repositoryDenison Resource Commonsen_US
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