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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 ]
 Graham Cooper Posts: 4,495 Registered: 5/20/10
Re: Can addition be defined in terms of multiplication?
Posted: Aug 17, 2013 7:06 PM

On Friday, August 16, 2013 1:54:40 AM UTC-7, 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 =,
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> S, +, *, and punctuation marks;
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>
>
> or the above with < as an additional binary predicate symbol.
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>
>
> --
>
> 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

Let's cheat a little and use a multiplication table...

Let's call this ummm... 'M'

M
--------

1 X 1 = 1
1 X 2 = 2
1 X 3 = 3
..

2 X 1 = 2
2 X 2 = 4
2 X 3 = 6
..

M states which rules are to be used.

Can..

(x+1) X y = (z+y) <-> x X y = z

(x+2) X y = (z+2Xy) <-> x X y = z

...

cover all the bases?

Herc
--
www.tinyurl.com/How-Prolog-Works

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