Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site pucc-i Path: utzoo!watmath!clyde!burl!ulysses!mhuxl!houxm!ihnp4!inuxc!pur-ee!CS-Mordred!Pucc-H:Pucc-I:ags From: ags@pucc-i (Seaman) Newsgroups: net.math Subject: Re: Pedantic Question - (nf) Message-ID: <215@pucc-i> Date: Wed, 22-Feb-84 11:17:00 EST Article-I.D.: pucc-i.215 Posted: Wed Feb 22 11:17:00 1984 Date-Received: Fri, 24-Feb-84 00:25:27 EST References: <5777@uiucdcs.UUCP> Organization: Purdue University Computing Center Lines: 79 > The definition of Peano numbers seems to be somewhat circular and not really > conclusive. Where is the leap where it states that strings of digits must be > finite length? > > Or perhaps by "subset," you mean "strict subset" and not "subset or the entire > set?" The Peano axioms are concerned with three primitive concepts: zero, number, and successor. To say that these three concepts are "primitive" means that we do not attempt to define what they mean. In fact, it is possible to interpret these concepts in more than one way. All that is required is that whatever meaning we eventually attach to the three primitive concepts must be consistent with the axioms. This is how circularity is avoided in mathematics: there are some things we simply do not define. There is nothing in the axioms about the representation of numbers in terms of digits. It is fairly easy to show that a model for the integers can be constructed by using finite digit strings to stand for "numbers", with the appropriate meanings of "zero" and "successor". In particular, "2" stands for "the successor of the successor of zero," etc. You can even build different models for the different number bases. Every time you change the underlying base, you change the meanings of "number" and "successor". In base ten, "101" is a number whose successor is "102". In base two, "101" is an entirely different number, whose successor is "110"; "102" is not a number at all. Both number systems satisfy the Peano axioms and are equally good models for the natural numbers. The "leap" which states that strings of digits must be of finite length depends only on the following assertion: "Let K be a digit string of finite length. Then the successor of K is also a digit string of finite length." This assertion is certainly true of the conventional representations of integers, using any base N. Look again at the fifth Peano axiom: (5) Let S be a subset of the numbers which obeys the rules: (a) Zero is in S. (b) The successor of any number in S is also in S. Then S contains all of the natural numbers. The finite digit strings represent a subset S which satisfies properties (a) and (b). The axiom says that there is nothing else -- S contains all of the natural numbers. Therefore no infinite digit strings can possibly represent natural numbers in a model of this type (i.e. the conventional ways of representing numbers in various bases). If you like, it is perfectly possible to model the numbers in a different way, so that infinite digit strings are possible. For example: (1) Zero is defined to be the string "000...", consisting of an infinite number of zeros. (2) A "number" is any string of the form "[111...1]000...", consisting of a finite string of ones (possibly empty), followed by an infinite string of zeros. (3) The "successor" of "[111...1]000..." is "111...1]1000...", where the number of ones has been increased by one. It is easy to see that these interpretations of "zero," "number," and "successor" are consistent with the Peano axioms, and therefore we have another model for the natural numbers. By the way, I used terms like "zero" and "one" in describing the model. This is not a circular definition. In fact, it is not a "definition" at all. I am merely using our conventional meanings of "zero" and "one" to describe a new model, then showing that the model is consistent with the Peano axioms. -- Dave Seaman ..!pur-ee!pucc-i:ags "Against people who give vent to their loquacity by extraneous bombastic circumlocution."