I agree with you. Actually you can use the second recursion theorem
of Kleene to collapse all the orders. This is easier in an untyped
programming language like (pure) LISP than in a typed language,
although some typed language have a primitive for handling untyped
self-reference, like the primitive SELF in Smalltalk ...

Bruno



At 23:29 19/01/04 -0800, Eric Hawthorne wrote:
How would they ever know that I wonder?
"Well let's see. I'm conscious and I'm not fallible. Therefore...." ;-)

David Barrett-Lennard wrote:

I'm wondering whether the following demonstrates that a computer that can
only generate "thoughts" which are sentences derivable from some underlying
axioms (and therefore can only generate "true" thoughts) is unable to think.

This is based on the fact that a formal system can't understand sentences
written down within that formal system (forgive me if I've worded this
badly).

Somehow we would need to support free parameters within quoted expressions.
Eg to specify the rule

It is a good idea to simplify "x+0" to "x"

It is not clear that language reflection can be supported in a completely
general way.  If it can, does this eliminate the need for a meta-language?
How does this relate to the claim above?

- David

I don't see the problem with representing logical meta-language, and meta-metalanguage... etc if necessary
in a computer. It's a bit tricky to get the semantics to work out correctly, I think, but there's nothing
"extra-computational" about doing higher-order theorem proving.


http://www.cl.cam.ac.uk/Research/HVG/HOL/

This is an example of an interactive (i.e. partly human-steered) higher-order thereom prover.
I think with enough work someone could get one of these kind of systems doing some useful higher-order
logic reasoning on its own, for certain kinds of problem domains anyway.


Eric



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