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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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 fom Posts: 1,968 Registered: 12/4/12
Re: Can addition be defined in terms of multiplication?
Posted: Aug 17, 2013 1:49 PM

On 8/17/2013 9:56 AM, Nam Nguyen wrote:
> On 17/08/2013 1:06 AM, Peter Percival wrote:
>> Nam Nguyen wrote:
>>> On 16/08/2013 9:49 AM, dullrich@sprynet.com wrote:
>>>> On Fri, 16 Aug 2013 02:05:09 -0700, William Elliot <marsh@panix.com>
>>>> wrote:

>>
>>>>> x + y = log(e^x * e^y)
>>>>
>>>> If you don't know what "in the language of arithmetic" means
>>>> it would be a good idea to refrain from answering questions
>>>> about the language of arithmetic, lest you look silly.

>>>
>>> _That_ actually isn't silly. The language of arithmetic is either
>>> L1(0,S,+,*,<) or L2(0,S,+,*), so "in the language of arithmetic"
>>> technically just means "in L1" or "in L2".

>>
>> So given the language of arithmetic is what you say, it isn't silly to
>> use log and exp?

>
> No. Take the "language of arithmetic" to be L1 (for example): you can
> formalize a real number system for log and exp.
>
> My point is it's silly to claim L1 to be the "language of arithmetic"
> while it could also be the "the language of complete order field", or
> many ... many other alternatives.
>

That may be so. However, the notion arises from universal algebra.

What one actually has is

"The theory of arithmetic"

"The language of the theory of arithmetic"

"The model of the language of the theory of arithmetic"

Tarski's 1933 specifically excludes consideration of
scientific languages built up from definition.

Apparently, Hilbert and Bernays demonstrated the eliminability
of definite descriptions (the syntax associated with defined
individual constants). Presuming that Kleene has represented
the argument faithfully, the requirements for obtaining reducts
can be found in "Introduction to Metamathematics". There
is a provability requirement.

When a theory is given axiomatically, its language is the alphabet
of symbols which are neither punctuation nor logical constants.

It is merely a list of symbols.

It is typed according to a list of ordered pairs which associate
an arity with each of the symbols.

It is significant that Chang and Keisler describe model theory
as

(universal algebra) + (logic)

to express the idea that algebraic structures which focus
primarily on operations (functions) and constants are extended
to include relations.

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