On Wed, Aug 22, 2012 at 10:48 AM, benjayk

> Bruno Marchal wrote:
> >
> >>
> >> Imagine a computer without an output. Now, if we look at what the
> >> computer
> >> is doing, we can not infer what it is actually doing in terms of
> >> high-level
> >> activity, because this is just defined at the output/input. For
> >> example, no
> >> video exists in the computer - the data of the video could be other
> >> data as
> >> well. We would indeed just find computation.
> >> At the level of the chip, notions like definition, proving, inductive
> >> interference don't exist. And if we believe the church-turing
> >> thesis, they
> >> can't exist in any computation (since all are equivalent to a
> >> computation of
> >> a turing computer, which doesn't have those notions), they would be
> >> merely
> >> labels that we use in our programming language.
> >
> > All computers are equivalent with respect to computability. This does
> > not entail that all computers are equivalent to respect of
> > provability. Indeed the PA machines proves much more than the RA
> > machines. The ZF machine proves much more than the PA machines. But
> > they do prove in the operational meaning of the term. They actually
> > give proof of statements. Like you can say that a computer can play
> > chess.
> > Computability is closed for the diagonal procedure, but not
> > provability, game, definability, etc.
> >
> OK, this makes sense.
> In any case, the problem still exists, though it may not be enough to say
> that the answer to the statement is not computable. The original form still
> holds (saying "solely using a computer").
For to work, as Godel did, you need to perfectly define the elements in the
sentence using a formal language like mathematics.  English is too
ambiguous.  If you try perfectly define what you mean by computer, in a
formal way, you may find that you have trouble coming up with a definition
that includes computers, but does't also include human brains.


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