Bruno Marchal wrote:
>> Imagine a computer without an output. Now, if we look at what the
>> is doing, we can not infer what it is actually doing in terms of
>> 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
>> 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
> 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").
Of course one can object to this, too, since it is not possible to solely
use a computer. We always use our brains to interpret the results the
computer gives us.
But its still practically true.
Just do the experiment and try to solve the question by programming a
computer. You will not be able to make sense of the question. As soon as you
cease to try to achieve a solution using the computer you will suddenly
realize the answer is YES since you didn't achieve a solution using the
computer (and this is what the sentence says).
The only way to avoid the problem is to hardcode the fact 'This statement
can't be confirmed to be true by utilizing a computer'=true into the
computer and claim that this a confirmation. But it seems that this is not
what we really mean by confirming, since we could program 'This statement
can't be confirmed to be true by utilizing a computer'=false into the
computer as well. It would just be a belief, not an actual confirmation.
Bruno Marchal wrote:
>> just because it can be
>> represented using computation. But ultimately a simple machine can't
>> the same as a complex one, because we need a next layer to interpret
>> simple computations as complex ones (which is possible). That is,
>> isn't as powerful as C++, because we need additional layers to
>> retrieve the
>> same information from the output of the assembler.
> That depends how you implement C++. It is not relevant. We might
> directly translate C++ in the physical layer, and emulate some
> assembler in the C++.
> But assembler and C++ are computationally equivalent because their
> programs exhaust the computable function by a Turing universal machine.
I think this is just a matter of how we define computation. If computation
is defined as what an universal Turing machine does, of course nothing can
be more computationally powerful.
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