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: 1,299
Registered: 10/25/10
Re: Can addition be defined in terms of multiplication?
Posted: Aug 18, 2013 8:48 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Ben Bacarisse wrote:
> William Elliot <marsh@panix.com> writes:
>

>> On Sun, 18 Aug 2013, Peter Percival wrote:
>>>>
>>>>>>> Can addition be defined in terms of multiplication? I.e.,
>>>>>>> is there a formula in the language of arithmetic
>>>>>>> x + y = z <-> ...
>>>>>>>
>>>>>>> such that in '...' any of the symbols of arithmetic
>>>>>>> except + may occur?
>>>>>>>
>>>>>>> The symbols of arithmetic (for the purpose of this question) are
>>>>>>> either
>>>>>>> individual variables, (classical) logical constants including =,
>>>>>>> S, +, *, and punctuation marks;
>>>>>>> or the above with < as an additional binary predicate symbol.

>>>>>>
>>>>>> How about
>>>>>> x + y = z <-> 2^x * 2^y = 2^z
>>>>>>
>>>>>> where 2^x is just an abbreviation for the function 2pwr: N -> N,
>>>>>> defined by
>>>>>> 2pwr(0) = 1
>>>>>> 2pwr( Sx ) = 2 * 2pwr( x )

>>>>> That goes beyond what I defined as the language of arithmetic.
>>>>
>>>> It does not. It quite definable with Peano's axioms
>>>> which may be presumed to be what you intend because
>>>> of the inclusion of S in the symbols of arithematic.

>>>
>>> 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.


Yes, I am. Sorry for not mentioning it. The definitions that I seek
will be eliminable.


--
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.