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

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 David C. Ullrich Posts: 3,531 Registered: 12/13/04
Re: Can addition be defined in terms of multiplication?
Posted: Aug 18, 2013 9:56 AM

On Sun, 18 Aug 2013 02:18:14 -0700, William Elliot <marsh@panix.com>
wrote:

>> 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 does.

>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.

You simply don't know what "a definition in the language
of arithmetic" means. But don't let that stop you.

>
>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