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, firstname.lastname@example.org wrote: >>>> On Fri, 16 Aug 2013 02:05:09 -0700, William Elliot <email@example.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.