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Topic: Can addition be defined in terms of multiplication?
Replies: 58   Last Post: Aug 23, 2013 3:56 PM

 Messages: [ Previous | Next ]
 Alan Smaill Posts: 1,018 Registered: 1/29/05
Re: Can addition be defined in terms of multiplication?
Posted: Aug 19, 2013 7:23 AM

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 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.

>
> Stepping out of my comfort zone here, but I think the point is that
> allowing recursive definitions makes the theory second-order, and raises
> the question of why one would not simply define + directly that way too.
>
> Broadly speaking, you can either have a second-order theory in which +
> and * and so on are not in the signature of the language (but are
> defined recursively) or you can have a first-order 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 first-order 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.

--
Alan Smaill

Date Subject Author
8/16/13 Peter Percival
8/16/13 William Elliot
8/16/13 Peter Percival
8/16/13 David C. Ullrich
8/16/13 namducnguyen
8/17/13 Peter Percival
8/17/13 namducnguyen
8/17/13 fom
8/23/13 tommy1729_
8/16/13 Peter Percival
8/16/13 Robin Chapman
8/16/13 Helmut Richter
8/16/13 Rotwang
8/16/13 Virgil
8/22/13 Rock Brentwood
8/16/13 Shmuel (Seymour J.) Metz
8/17/13 Helmut Richter
8/16/13 Jim Burns
8/16/13 fom
8/17/13 Robin Chapman
8/17/13 fom
8/17/13 Peter Percival
8/17/13 fom
8/17/13 Peter Percival
8/17/13 Peter Percival
8/18/13 William Elliot
8/18/13 Peter Percival
8/18/13 William Elliot
8/18/13 Peter Percival
8/18/13 Graham Cooper
8/18/13 David C. Ullrich
8/18/13 David C. Ullrich
8/17/13 Graham Cooper
8/18/13 David Bernier
8/18/13 Ben Bacarisse
8/18/13 Peter Percival
8/18/13 Jim Burns
8/18/13 fom
8/18/13 Ben Bacarisse
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/18/13 Graham Cooper
8/19/13 Graham Cooper
8/19/13 Alan Smaill
8/19/13 fom
8/19/13 Alan Smaill
8/20/13 Alan Smaill
8/20/13 Peter Percival
8/20/13 Graham Cooper
8/20/13 Graham Cooper
8/22/13 David Libert
8/22/13 Peter Percival
8/20/13 fom