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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 ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Can addition be defined in terms of multiplication?
Posted: Aug 16, 2013 9:21 PM

On 8/16/2013 7:05 PM, 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 )
>
> It's true that I use the successor S, which I hear implicitly
> uses addition, but without S I don't see how to do much of
> anything at all.
>
> I wonder how Nam defines * without using either + or S.
>
> Come to think of it, if I can use S, why can't I just
> go ahead and define + in the usual (so far as I know) fashion
> x + 0 = x
>
> x + Sy = S(x + y)
>
> It seems to me that, if one is going to do arithmetic
> without addition, unique prime factorization becomes central.

I have not really developed my ideas in this area much. However,
while trying to consider certain alternative interpretations for
some sentences in which I had been interested, I came up with the
following:

--------------------------------

AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

may be interpreted as "x is a proper divisor of y"
and

AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

may be interpreted as "x is a prime
divisor of y".

--------------------------------

The idea here, of course, is to begin with respect to aliquot
parts (a proper part relation is irreflexive and transitive).

Because of relatively simple analogous constructions discussed
in Ian Fleming's "Combinatorics of Finite Sets" the only model for
these sentences which I have considered are the natural numbers
with no squares (powers) in their prime decompositions.

The sentence which distinguishes interpretation of the predicates
is

AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) -> Az((xez /\ -ycz) ->
(-(Ew(wey /\ -xew) <-> Aw(xcw <-> ycw)))))

or, more simply,

AxAy(xey -> Az((xez /\ -ycz) -> (-(Ew(wey /\ -xew) <-> Aw(xcw <-> ycw)))))

respect to a very simple model for the purpose of alternative
interpretation.

The material in Fleming's book that influenced the restriction
on the model is the fact that the poset of subsets of an n-set
can be considered as the poset of divisors of square-free numbers
under division. So, from the point of view of set theory,
this would be the relevant case for comparison.

I have begun thinking about developing this for a theory of
arithmetic. But, I am not an arithmetician. Heck, I am not
even a mathematician. :-)

Beginning with divisors is a bit different from what you had
in mind. But, given your statement, I thought I would throw
this into the mix.

<snip>

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