Re: Penrose vs the Robot


Daryl McCullough wrote:
<snip>
>
> I claim that Penrose *cannot* produce such a statement. Penrose
> suggests that S can be constructed as follows:
>
> Let T be the set of all statements in the language of Peano
> arithmetic that the robot considers unassailably true. Then
> since the robot's brain is a Turing machine, T is an r.e.
> set, so T is actually definable in PA. Then we can let S
> be the sentence "T is consistent". If T is indeed a consistent
> extention of PA, then by Godel's theorem, S is not provable
> by T.
>
> But I claim that there *is* no such theory T, and so there
> is no such statement S. The existence of such a theory T
> does not follow from the fact that the robot has a Turing
> machine brain.

Unless you are claiming that "considers unassailably true" is
meaningless for a robot (a claim I would have some sympathy for,
whether for a robot or humans), I don't see how you can claim there is
no such theory T. T is just a set of statements which is (1) a subset
of the set of all stements in PA; and (2) whose elements satisfy the
condition of being unassailably true for the robot. So, again unless
your claim is that there is some problem with "unassailably true", by
Separation T exists. Or am I missing some subtlety here?

.

Relevant Pages

  • Re: Penroses reply to Chalmers
    ... >being unassailably true to it being unassailably true. ... >work for the robot. ... >I think what you need to be disputing is that Penrose really does know ... Show that leads to a contradiction via ...
    (sci.logic)
  • Re: Penroses reply to Chalmers
    ... >>being unassailably true to it being unassailably true. ... >>work for the robot. ... seems to be a weak point that Penrose can argue against. ... > Penrose is not arguing that he is formal system F, ...
    (sci.logic)
  • Re: Penroses reply to Chalmers
    ... >>being unassailably true to it being unassailably true. ... >>work for the robot. ... Penrose claims his axioms ... Then some of the axioms are false. ...
    (sci.logic)
  • Re: Penroses reply to Chalmers
    ... >>are the axioms of PA. ... > That's all that is necessary for the robot to carry out Penrose' argument. ... being unassailably true to it being unassailably true. ...
    (sci.logic)
  • Re: Penrose vs the Robot
    ... > p is a simulation of Penrose. ... > on *how* the robot came to find out that it is p. ... > Human beings do not work by deduction from axioms, ... "Human beings do not work by deduction from axioms" ...
    (sci.logic)