Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Peter Percival

Posts: 940
Registered: 10/25/10
Re: Can addition be defined in terms of multiplication?
Posted: Aug 20, 2013 4:53 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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


Which led me to
http://math.stackexchange.com/questions/449146/why-are-addition-and-multiplication-included-in-the-signature-of-first-order-pea?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


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

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.