On 3/9/2014 7:16 PM, Virgil wrote: > In article <email@example.com>, > WM <firstname.lastname@example.org> wrote: > >> Here we are concerned with the question whether the Peano axioms define the >> natural numbers. > > They certainly allow a definition of the natural numbers which > definition is impossible if one rejects them. >
"Rejects" is not the same as "finds immaterial", that multiplication is "defined" well enough in terms of repeated addition, from arithmetic that is less than Peano arithmetic (+,*). [One need not reject them to ignore them.]
Would you agree that Peano axioms E 0 and f.e. E x that E S(x), this is addition and multiplication? Here this is to separate the definitions of successorship, by itself, from addition and multiplication, as themselves defined. Peano arithmetic is (N, +, *) compared to that the Peano axioms as above is (N), with no defined arithmetic other than that it is categorical. To have then that Peano arithmetic and algebra follows from Peano axioms then is as to whether here for example otherwise there are space terms, that successorship defines the space of the arithmetic, and algebra.
The Peano axioms "exists 0" and "for each X, exists S(X)", here has whether "S(X) is the unique successor of X" or "S(X) is X+1", that that the second has the requirement of definition for addition, "+", also "1". Then, a definition of addition is sufficient to implement multiplication. It seems also a definition of multiplication would be enough to carry out addition, though it might see division defined first.