
Re: Can addition be defined in terms of multiplication?
Posted:
Aug 20, 2013 4:53 AM


Alan Smaill wrote: > fom <fomJUNK@nyms.net> writes: > >> On 8/19/2013 6:23 AM, Alan Smaill wrote: >>> Ben Bacarisse <ben.usenet@bsb.me.uk> writes: >>> >>>> William Elliot <marsh@panix.com> writes: >>>> >>>>> 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. >>> >>> I do too; it can be done, but it is not easy. >>> >>> See Goedel on defining exponentiation from plus and times via the >>> Chinese remainder theorem. >>> >> >> I have several volumes of the complete works. >> >> Do you have any more specific information on >> which paper? > > There is a formulation in the incompleteness theorem article. > he needed it to know that goedel numbers using exponentiation could > be defined inside arithmetic with plus and times. > > Versions of this use just FOL; > overview here: > > http://math.stackexchange.com/questions/312891/howisexponentiationdefinedinpeanoarithmetic
Which led me to http://math.stackexchange.com/questions/449146/whyareadditionandmultiplicationincludedinthesignatureoffirstorderpea?rq=1 near the foot of which it says:
Actually, more is known. Neither addition nor multiplication is definable from successor alone; multiplication is not definable from successor and addition; and addition is not definable from successor and multiplication. The theory of the natural numbers with multiplication and addition is undecidable, but the restriction to just addition is decidable, and the restriction with just multiplication is decidable.
 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

