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

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 Ben Bacarisse Posts: 1,972 Registered: 7/4/07
Re: Can addition be defined in terms of multiplication?
Posted: Aug 18, 2013 11:07 AM
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Jim Burns <burns.87@osu.edu> writes:

> On 8/18/2013 8:02 AM, 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.
>>
>> <snip>
>>

>
> Does "first order theory" mean no recursive definitions or
> no recursion at all?

No, but it means a weaker induction axiom than the one usually used in a
second-order theory. At this point you bump up hard against my
knowledge boundary and I can't really say much more about it.

I should pass on the rest. Someone Who Knows will come along clear it
up, I hope.

<snip>
--
Ben.

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

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