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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 6:09 AM

On Sun, 18 Aug 2013, 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.

> > > >
> > > > How about
> > > > 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.

>
> Then I think the onus is on you to produced definitions in one or both of
> these forms:
> x + y = ...
> x + y = z <-> ...
>
> where the only non-logical symbols (baring punctuation) in the ... are from
> this set: {*,S,0} or this set: {*,S,0,<}. I wouldn't be surprised if + can be
> defined (in the way requested) from {*,S,0} or {*,S,0,<} but I would like
> either to see it spelt out, or to be given a reference.

As Jim Burns said
z = x + y iff 2^z = 2^x * 2^y

where 2^n is defined by induction 2^0 = 1, 2^1 = 1 and 2^(n+1) = 2*2^n
all of which can be done with Peano's axioms.

> > If you want it for the reals,
>
> Which I don't.
>

> > 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