
Re: Can addition be defined in terms of multiplication?
Posted:
Aug 18, 2013 8:48 AM


Ben Bacarisse wrote: > William Elliot <marsh@panix.com> writes: > >> On Sun, 18 Aug 2013, 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. >>>>>> >>>>>> 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 ) >>>>> 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 nonlogical 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. >> >> As Jim Burns said >> z = x + y iff 2^z = 2^x * 2^y >> >> where 2^n is defined by induction 2^0 = 1, 2^1 = 1 and 2^(n+1) = 2*2^n >> all of which can be done with Peano's axioms. > > Stepping out of my comfort zone here, but I think the point is that > allowing recursive definitions makes the theory secondorder, and raises > the question of why one would not simply define + directly that way too. > > Broadly speaking, you can either have a secondorder theory in which + > and * and so on are not in the signature of the language (but are > defined recursively) or you can have a firstorder theory where + and * > and so on are added to the signature, with axioms used to induce the > usual meaning. > > I suspect Peter is talking about a firstorder theory where recursive > definitions are not permitted.
Yes, I am. Sorry for not mentioning it. The definitions that I seek will be eliminable.
 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

