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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 ]
 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Can addition be defined in terms of multiplication?
Posted: Aug 18, 2013 5:18 AM

> Jim Burns wrote:
> > On 8/16/2013 4:54 AM, 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?
> > >
> > > 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.

> >
> > x + y = z <-> 2^x * 2^y = 2^z
> >
> > where 2^x is just an abbreviation for the function 2pwr: N -> N,
> > defined by
> > 2pwr(0) = 1
> > 2pwr( Sx ) = 2 * 2pwr( x )

> That goes beyond what I defined as the language of arithmetic.

It does not. It quite definable with Peano's axioms
which may be presumed to be what you intend because
of the inclusion of S in the symbols of arithematic.

If you want it for the reals, then 2^x, x in is
definable with <= and a whole lot of logical overhand.

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