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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 18, 2013 5:31 AM

William Elliot 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 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.

>
> 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.
>

--
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