Date: Aug 18, 2013 6:17 AM
Author: Peter Percival
Subject: Re: Can addition be defined in terms of multiplication?

William Elliot wrote:
> 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.


And the magic formulae

2^x = ...

2^x = y <-> ...

are what in the languages {*,S,0} or {*,S,0,<}?

>>> If you want it for the reals,
>>
>> Which I don't.
>>

>>> then 2^x, x in is
>>> definable with <= and a whole lot of logical overhand.

>


--
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