Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


quasi
Posts:
12,018
Registered:
7/15/05


Re: Induction is Wrong
Posted:
Nov 30, 2012 3:37 AM


RussellE wrote:
>Andrew Boucher has developed a theory called >General Arithmetic (GA): ><http://www.andrewboucher.com/papers/ga.pdf> > >GA is a subtheory of Peano Arithmetic (PA). >If we add an induction axiom (IND) to the axioms of >Ring Theory (RT) then GA is also a subtheory of RT+IND. >(We also need a weak successor axiom).
But GA is not a subtheory 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 noncommutative rings. >There are even finite noncommutative 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 noncommutative ring is a model of GA. Similarly, no commutative ring (such as Z) which fails to satisfy the 4squares theorem is a model of GA.
>Russell >Integers are an illusion
It's your claimed contradiction  that's the illusion.
quasi



