[hist-analytic] Methods of Proof: Re: Clarity Is Not Enough
Roger Bishop Jones
rbj at rbjones.com
Mon Feb 23 11:24:00 EST 2009
On Monday 23 February 2009 16:02:34 steve bayne wrote:
> Now on arithmetization of
> syntax, one would think that the Godel numbers occur
> in the meta-language.
Yes, that's right.
> Making explicit the ontology of arithmetization would, then,
> seem to require a yet higher order language.
I'm not clear what you mean here by "making explicit the ontology".
> "In general, to reason about language A in language B one must
> be able to represent the syntax of A in the ontology of B."
This is what we are talking about here.
There are just two languages involved, A and B.
To talk about the syntax of A in the language B we have to
have available in B objects which will suffice as representatives
of the syntactic objects of A.
B is our metalanguage and A is the object language.
If B is arithmetic then we have available the
natural numbers and can use Godel numbers to talk about the
syntax of A. To talk about B in A we don't need any further
metalanguage, If we wanted to formalise this discussion
then we would need another metalangage, for a transparent and
convincing formalisation set theory would probably be best.
Roger
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