Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Can addition be defined in terms of multiplication?
Replies: 58   Last Post: Aug 23, 2013 3:56 PM

 Messages: [ Previous | Next ]
 Helmut Richter Posts: 164 Registered: 7/4/06
Re: Can addition be defined in terms of multiplication?
Posted: Aug 16, 2013 12:14 PM

On Fri, 16 Aug 2013, Robin Chapman wrote:

> On 16/08/2013 09:54, Peter Percival wrote:
> > Can addition be defined in terms of multiplication? I.e., is there a
> > formula in the language of arithmetic
> >
> > x + y = z <-> ...
> >
> > such that in '...' any of the symbols of arithmetic except + may occur?
> > Or, alternatively, is there a formula in the language of arithmetic
> >
> > x + y = ...
> >
> > with the same requirement?

A similar problem I have asked some years ago is the following:

Given a multiplication on a set (e.g. defined as a commutative and
associative operation allowing cancellation (ab = ac implies b = c)),
is there an addition so that the set becomes a ring with both operations?
I have no clue how to tackle such questions.

An example: Let M = {x elem Z : x == 1 mod 3} with ordinary
multiplication. Could this be the multiplication in a ring, if addition is
suitably defined? I guess, no, but it is but a guess.

> IIRC "arithmetic without addition" is a decidable theory
> (of course Presburger's famous theorem is that "arithmetic
> without multiplication" is decidable). I think this means that
> the theory of the language (0,1,*,=) in N is decidable. I'm not
> sure if we can allow S and/or <.
>
> Of course if we have a language like this with a decidable theory, then
> we can't define addition, lest the full theory of Peano arithmetic
> be decidable.

I'll look into this argument a bit closer.

--
Helmut Richter

Date Subject Author
8/16/13 Peter Percival
8/16/13 William Elliot
8/16/13 Peter Percival
8/16/13 David C. Ullrich
8/16/13 namducnguyen
8/17/13 Peter Percival
8/17/13 namducnguyen
8/17/13 fom
8/23/13 tommy1729_
8/16/13 Peter Percival
8/16/13 Robin Chapman
8/16/13 Helmut Richter
8/16/13 Rotwang
8/16/13 Virgil
8/22/13 Rock Brentwood
8/16/13 Shmuel (Seymour J.) Metz
8/17/13 Helmut Richter
8/16/13 Jim Burns
8/16/13 fom
8/17/13 Robin Chapman
8/17/13 fom
8/17/13 Peter Percival
8/17/13 fom
8/17/13 Peter Percival
8/17/13 Peter Percival
8/18/13 William Elliot
8/18/13 Peter Percival
8/18/13 William Elliot
8/18/13 Peter Percival
8/18/13 Graham Cooper
8/18/13 David C. Ullrich
8/18/13 David C. Ullrich
8/17/13 Graham Cooper
8/18/13 David Bernier
8/18/13 Ben Bacarisse
8/18/13 Peter Percival
8/18/13 Jim Burns
8/18/13 fom
8/18/13 Ben Bacarisse
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/19/13 Graham Cooper
8/19/13 Alan Smaill
8/19/13 fom
8/19/13 Alan Smaill
8/20/13 Alan Smaill
8/20/13 Peter Percival
8/20/13 Graham Cooper
8/20/13 Graham Cooper
8/22/13 David Libert
8/22/13 Peter Percival
8/20/13 fom