[hist-analytic] From Richard Grandy: What do we need to represent syntax?
steve bayne
baynesrb at yahoo.com
Tue Feb 24 07:58:57 EST 2009
--- On Mon, 2/23/09, Richard Grandy <rgrandy at rice.edu> wrote:
From: Richard Grandy <rgrandy at rice.edu>
Subject: What do we need to represent syntax?
To: baynesrb at yahoo.com, "Roger Bishop Jones" <rbj at rbjones.com>
Date: Monday, February 23, 2009, 7:34 PM
What do we need to represent
syntax?It may be natural or habitual to think about ontology or
domains of discourse in this context, but if we are analyzing what
is required we need to think more carefully.
Godel's theorem for any specific system is strictly a proof
theoretic result, no models or domains required, thank
you. "If S is omega consistent neither G or ~G is provable
in S"
Godel's generalized theorem ("For any formal system S
....") requires recursion theory or something equivalent to
give a precise definition of "formal". Again no models
or domains required.
I know that Godel's theorem was probably not what Steve had in
mind, but is the crispest example, and I don't see offhand why
more is needed for his purposes (e.g., measuring redundancy).
To put it more directly, I am arguing that what is required for a
metalanguage M to provide resources to analyze the syntax of language
L is that the syntax of M can represent the syntax of L.
Richard
Raised by proof-theoreticians and recursion theorists on the East
Coast, though later persuaded (during time on the sunny West
Coast) that model theory has its virtues)
Yes, I think talking in terms of domains
of discourse
is the right way to go. Now on arithmetization of
syntax, one would think that the Godel numbers occur
in the meta-language. Making explicit the ontology
of arithmetization would, then, seem to require a
yet higher order language. What we are talking about
in the case of the formalization of arithmetic is
wff in the object language and these seem to contain
few "Godel numbers." Note that being a prime number
is part of how we assign the Godel numbers and so
there may an implicit "ontology" of arithmetic
involved. I better shut up. I haven't done much
logic in a lot of years.
Regards
Steve
--- On Mon, 2/23/09, Roger Bishop Jones <rbj at rbjones.com>
wrote:
> From: Roger Bishop Jones
<rbj at rbjones.com>
> Subject: Re: Methods of Proof: Re: Clarity Is Not Enough
> To: baynesrb at yahoo.com
> Cc: hist-analytic at simplelists.com
> Date: Monday, February 23, 2009, 10:56 AM
> On Monday 23 February 2009 15:27:26 steve bayne wrote:
> > I'm pretty sure I get what you are talking about
> w.r.t
> > the metalanguage stuff. But take this fragment:
> >
> > "represent the syntax of A in the ontology of
> B."
> >
> > Could you give an example of syntax being represented
> > "in the ontology"?
>
> Take arithmetisation.
>
> The arithmetisation of syntax consists in assigning to
> each syntactic entity a natural number so that talk
> about syntax can be translated into talk about numbers
> (and partially vice-versa).
>
> The ontology of arithmetic is the natural numbers.
> Arithmetisation provides numerical representatives
> for syntactic entities, and hence represents syntax
> in the ontology of arithmetic.
>
> It would perhaps have been clearer to talk of
> "the domain of discourse of B", but that is of
> course just the set of things which exist so far
> as B is concerned, i.e. "the ontology of B".
> (is this an odd usage of "ontology"?
> I have thought about foundations "ontologically"
> for so long that I may have become accustomed to
> a way of thinking and hence writing which is not common)
>
> Roger
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