fom
Posts:
1,968
Registered:
12/4/12


Re: Can addition be defined in terms of multiplication?
Posted:
Aug 20, 2013 5:51 AM


On 8/20/2013 3:38 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 >
thanks

