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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 ]
 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 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.
>>>

>>
>> 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/how-is-exponentiation-defined-in-peano-arithmetic
>

thanks

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