Nam Nguyen wrote: > > On 13/04/2013 7:10 PM, Jesse F. Hughes wrote: > > Nam Nguyen <email@example.com> writes: > > > >> On 13/04/2013 3:18 PM, Peter Percival wrote: > >>> Nam Nguyen wrote: > >> > >>> > >>> No, I know of more than one signature for PA (and I wouldn't be > >>> surprised if there are others) so I'm asking you which of them you are > >>> using. > >>> > >>>> Now that you understand which of L(S,+,x,0) and L(<,S,+,x,0) > >>> > >>> I don't know. > >> > >> OK. Let's cut the chase then: For L(PA) I've used what Shoenfield used, > >> which is L(<,S,+,x,0). > > > > I wonder why the heck you couldn't have said that several posts ago. > > You should not have wondered why the heck that was the case because: > > (a) iirc, for 10+ years around here, typical posters like Torkel > Franzen, Chris Menzel, Franz Fritsche, Aatu Koskensilta, > George Greene, Jesse F. Hughes, etc.. have not have to spell > out all the symbols of L(PA), all the axioms of PA, when they > mentioned _the familiar_ "L(PA)" or "PA" and there was no need > to spell them out. > > Why should I do anything different from what typical posters have > done in this respect.
They, bar one, can answer for themselves. Nevertheless I am going to guess at their response. If asked if '<' was in the signature of L(PA) they would say yes or no. They might also say, according to the circumstance, that it had no bearing on the discussion. But you can't say that: I refer you to a recent exchange between us that is reproduced below.
> (b) What is the need for anyone to know exactly what the signature > of _the familiar_ L(PA) that one has to bother asking people? > What's the difference in this context? Would a choice between > 2 versions of L(PA) that needs to be taken for granted hurt > any argument in this context? > > (c) I could have suggested to them to read standard textbooks and choose > at will which signature choice they'd prefer: in this context it > does _NOT_ matter what choice; and so one doesn't have to be > bothered with.
One textbook that you refer to is Shoenfield's. It is rather advanced for me, for no one knows less about logic than I, but in its early pages it makes clear that a structure's definition depends on the language that it is "about". (That other authors may introduce structure first and then language I don't doubt.) You claimed (see below) that the structure in question would include properties being even, being odd, being prime, etc. According to Shoenfield, a structure for L(PA) will only have those properties (= subsets of N) if L(PA) has corresponding predicates. (S. writes "predicate symbol" where I would have "predicate", and he writes "predicate" where I would have "property or relation". No matter.) So your claim would only be true if L(PA) had predicates, say E, O, P, corresponding to those properties. You seem not to understand the distinction between a symbol being in a language's signature and a symbol being definable in the language.
> Now that that has been spelled out, however unnecessarily, what's next?
Here is the post (dated Sat, 13 Apr 2013 17:58:45 +0100) referred to above: " Nam Nguyen wrote: > > On 13/04/2013 10:22 AM, Peter Percival wrote: > > Nam Nguyen wrote: > > > >> > >> I don't know what your point might have been there but it's important > >> that what you said wasn't quite correct given we're arguing the language > >> structure in which there are quite a few _important_ 1-ary (n < 2) > >> properties, such as being an even, odd, being a prime, etc... > > > > The structure for "the" language of PA has none of being an even, odd, > > being a prime, among its properties. > > That's not true: those properties are _definable_ using the language > symbols you're about to mention below.
The properties, relations, functions and distinguished elements in a language structure correspond one-to-one with the predicates, relation symbols, function symbols and individual constants in the language. What is definable doesn't come into it.
> > I put "the" in scare quotes > > because (I think) the most common first order language for PA has > > functions S, + and x, and distinguished element 0. In that case the > > only unary property is "being 0". But there is at least one other first > > order language for PA that I have come across, it has <. "