fom
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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. > > How about > > 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)))))
As I said, I have not really thought about this except with 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 nset can be considered as the poset of divisors of squarefree 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>

