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Topic: Induction is Wrong
Replies: 8   Last Post: Nov 30, 2012 5:55 PM

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 quasi Posts: 12,067 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 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:
>
>Induction is wrong. It proves multiplication,
>as defined by the axioms of ring theory,
>must be commutative when this is not true.

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

quasi

Date Subject Author
11/29/12 reasterly@gmail.com
11/29/12 J. Antonio Perez M.
11/29/12 Dan Christensen
11/30/12 quasi
11/30/12 Brian Q. Hutchings
11/30/12 Brian Q. Hutchings
11/30/12 quasi
11/30/12 Rupert
11/30/12 Brian Q. Hutchings