Peano's Postulates 1889 – In Original Italian(adsbygoogle = window.adsbygoogle || []).push({});

Nel 1889 pubblica Arithmetices Principia nova methodo exposita opera, tutta in latino, famosa in tutto il mondo: la teoria dei numeri naturali si sviluppa a partire da cinque semplici proprietà (gli assiomi di Peano):

I. Uno è un numero naturale

II. Per ogni numero naturale n esiste un unico numero naturale n* detto successore di n

III. Uno non è successore di alcun numero naturale

IV. Se x* = y*, allora x = y

V. Se K è una proprietà tale che:

Uno ha la proprietà K

per ogni k appartenente a N, se k ha la proprietà K, anche k* la la stessa proprietà

allora la proprietà K vale per tutti numeri naturali. (Principio di induzione).

http://www.webfract.it/FRATTALI/vitaPeano.htm#se

I do not see how quantity based on order can be derived from this. Let's take two examples:

Take a number of cows coming back from pasture. They all line up and follow one another in succession. A cow is a number, the successor of a cow is a number - a cow, and so forth. Every cow has a successor (save for the last - an exception to Peano, but not critical to this argument), and no two cows have the same successor cow. Yet, there is nothing establishing the concept of order based on size. We can wait for the concept of ordering by quantity until the cows come home. Well, lets say they did, but Peano says that these cows, each called “number”, the leading one called “zero”, must follow each other. Each cow is unique, and one must be behind the other. Two or more cows cannot follow directly behind the other, and each cow has one in front of her. Again, we have more than one cow, but the fact that one follows the other does not say anything about quantity. Or is it that succession, itself is adding? Yet I do not see how a particular number is attributed to a particular member standing in a line resulting from succession.

Second example:

Take three bottles, each of a different height, weight, and color, and set them on a table. One is asked to make one succeed, or follow, another. It can be readily demonstrated that based on Peano, that none of these properties has to be in any order - only that one bottle succeeds the other and several different orderings are possible. However, merely saying that each s unique and one must follow another says nothing about ordering in terms of ascending quantity.

Again, I may be missing something, but about the only way one can introduce a concept of quantity is by adding an axiom either from Zermelo-Fraenkel or von Neumann set theory.

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# Peano's Postulates - question

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