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

 Messages: [ Previous | Next ]
 Robin Chapman Posts: 412 Registered: 5/29/08
Re: Can addition be defined in terms of multiplication?
Posted: Aug 16, 2013 6:48 AM

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?
>
> The symbols of arithmetic (for the purpose of this question) are either
>
> individual variables, (classical) logical constants including =,
> S, +, *, and punctuation marks;
>
> or the above with < as an additional binary predicate symbol.

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.

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