Peano's Postulates 1889 – In Original Italian 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.
I am confused by your question. There is nothing about "size" or "quantity" in Peano's axioms. The notion of positive integers, and counting, telling us the "size" of a set cannot be derived from Peano's axioms.
Thank you for your reply HallsofIvy. This is precisely the point I see. Rosser (Logic for Mathematicians – 1953 – p. 279) states: “Actually, Peano stated his axioms for the positive integers, rather than for the negative integers. However, his axioms are just what the above five statements become if we interpret Nn as the class of positive integers, and replace 0 by 1 throughout. “ (It would take a month of Tuesdays to type in Rosser's theorems, but the key word here is “interpret”. To me, I thought that axioms had to be explicit. Suppes (Axiomatic Set Theory – 1970 – p. 122) states: ”It is a proper question and not an idle philosophical speculation to ask why mathematicians agree almost uniformly on Peano's axioms. Deep and difficult theorems about the natural numbers were proved before any adequate axioms were formulated. Putative axioms which did not yield these theorems would be rejected because , it seems, independent of any axioms there is quite a precise notion of what is true or false of the natural numbers. The author is not prepared to give any exact account of these intuitive notions... ... that the natural numbers are the finite cardinals or the finite ordinals is not to give such an exact account, for these intuitive yet precise ideas about the natural numbers are themselves are used in deciding if the proposed identification is acceptable.” It is my thinking that axioms are to be precise enough not to need interpretation. What you are saying confirms what I have thought for some time about the postulates not giving rise to a quantity based on ordering, or succession. What confuses me is that the postulates are referred to as being the foundations of mathematics, i.e., the number system, and a number systems is all about quantity and succession. This is why I stated that it would take a ZF, von Neumann, or other set theoretical basis in the form of axioms to add that idea of quantity based on succession. Succession, by itself, is not adequate for the foundation of a number system. If I am missing something here or going down some weird rabbit hole, please let me know. I could be missing something obvious.