fom
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12/4/12


Re: Can addition be defined in terms of multiplication?
Posted:
Aug 19, 2013 7:00 PM


On 8/19/2013 6:23 AM, Alan Smaill wrote: > Ben Bacarisse <ben.usenet@bsb.me.uk> writes: > >> William Elliot <marsh@panix.com> writes: >> >>> On Sun, 18 Aug 2013, Peter Percival wrote: > ... >>>> 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. > > 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?

