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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 ]
 Peter Percival Posts: 2,623 Registered: 10/25/10
Re: Can addition be defined in terms of multiplication?
Posted: Aug 17, 2013 10:29 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? 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.

>
>
> 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.

--
Sorrow in all lands, and grievous omens.
Great anger in the dragon of the hills,
And silent now the earth's green oracles
That will not speak again of innocence.
David Sutton -- Geomancies

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