>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).
But GA is not a sub-theory of RT.
>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.
Your logic is badly flawed.
The fact that every model of GA has commutative multiplication does not imply that every model of GA is a ring, nor that every ring is a model of GA. In fact, it's clear that most of the commonly encountered rings are not models of GA. For example, no non-commutative ring is a model of GA. Similarly, no commutative ring (such as Z) which fails to satisfy the 4-squares theorem is a model of GA.
>Russell >-Integers are an illusion
It's your claimed contradiction -- that's the illusion.