On Nov 29, 9:27 pm, RussellE <reaste...@gmail.com> wrote: > Andrew Boucher has developed a theory called General Arithmetic (GA):http://www.andrewboucher.com/papers/ga.pdf > > GA is a sub-theory of Peano Arithmetic (PA). > If we add an induction axiom (IND) to the axioms of Ring Theory (RT) > then > GA is also a sub-theory of RT+IND. (We also need a weak successor > axiom). > > Boucher proves Lagrange's four square theorem, every number is the sum > of four > squares, is a theorem of GA. Since the four square theorem is not true > in the > integers, the integers can not be a model for GA, PA, or RT+IND. > > GA also proves multiplication is commutative. > It is well known there are non-commutative rings. > There are even finite non-commutative rings:http://answers.yahoo.com/question/index?qid=20090827201012AAD7qJg > > Induction is wrong. It proves multiplication, > as defined by the axioms of ring theory, > must be commutative when this is not true. >
Wrong. The set of integers, along with usual addition and multiplication functions on the integers can be constructed starting from Peano's Axioms (including induction) by using the axioms of logic and set theory.