Date: Nov 30, 2012 12:44 PM Author: Frederick Williams Subject: Re: Induction is Wrong Dan Christensen wrote:

>

> On Nov 30, 10:34 am, Dan Christensen <Dan_Christen...@sympatico.ca>

> wrote:

> > On Nov 30, 12:35 am, Dan Christensen <Dan_Christen...@sympatico.ca>

> > wrote:

> >

> >

> >

> >

> >

> >

> >

> >

> >

> > > On Nov 29, 11:45 pm, reaste...@gmail.com wrote:

> >

> > > > On Thursday, November 29, 2012 7:59:32 PM UTC-8, Dan Christensen wrote:

> > > > > 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

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> > > > > > 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

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> > > > > > 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

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> > > > > > Induction is wrong. It proves multiplication,

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> > > > > > 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.

> >

> > > > > Dan

> >

> > > > > Download my DC Proof 2.0 software athttp://www.dcproof.com

> >

> > > > Yes, you can construct the integers in PA,

> > > > but the integers can not be the universe of a

> > > > model of PA.

> >

> > > > My point is that induction proves multiplication

> > > > must be commutative even when multiplication

> > > > is defined with the axioms of ring theory.

> > > > Yet, we know multiplication does not have to

> > > > be commutative.

> >

> > > Induction cannot be applied to rings in general if that is what you

> > > are getting at. That doesn't mean induction is "wrong."

> >

> > Hmmmm... I wonder if, within any infinite ring, there does exist an

> > infinite sub-ring that is indeed commutative. There does exist within

> > it, a smallest subset {1, 1+1, 1+1+1, ...}

>

> That should be the smallest subset containing 1, 1+1, 1+1+1, ...

For a ring you'll need -1, -(1+1), -(1+1+1), ... as well.

Do all your rings have a multiplicative neutral element?

--

When a true genius appears in the world, you may know him by

this sign, that the dunces are all in confederacy against him.

Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting