```Date: Aug 18, 2013 6:17 AM
Author: Peter Percival
Subject: Re: Can addition be defined in terms of multiplication?

William Elliot wrote:> 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 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.>> 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.And the magic formulae   2^x = ...   2^x = y <-> ...are what in the languages {*,S,0} or {*,S,0,<}?>>> 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 oraclesThat will not speak again of innocence.       David Sutton -- Geomancies
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