Re: .EXE -> .ASM -> .EXE


Charles A. Crayne wrote:
On 24 Jun 2006 18:47:11 -0700
"randyhyde@xxxxxxxxxxxxx" <randyhyde@xxxxxxxxxxxxx> wrote:

:this is a consequence of Godel's incompleteness theorem

I do wish that you would take the time to understand the this theorem,
instead of just waving it around like Van Helsing fending off Dracula with
a crucifix.

To begin with, Godel published his work in 1931, so it should be obvious
to the casual observer that he was not talking about artificial computers,
but rather real human beings, working within those formal logic systems
(such as arithmetic, algebra, geometry, and symbolic logic) which are the
basis of our modern technological society.

Well, then, since *you're* the expert, why not explain it?
Forgive me, but part of getting a graduate education included learning
about these types of things. I would be interested in seeing you
contradict what I've learned over the years.



Consider the following classical examples of undecidability:

Let (statement) A = "This statement is true".
Let (statement) B = "This statement is false".

No. At least not in the field of computer science. These examples are
trivially decidable. From Hopcroft and Ullman:

A problem whose language is recursive is said to be decidable.
Otherwise the problem is undecidable. That is, a problem is undecidable
if there is no algorithm that takes as input an instance of the problem
and determines whether the answer to that instance is yes or no.

An unintuitive consequence of the definition of undecidable is that
problems with only a single instance [and that includes your examples
above] are trivially decidable. ... There is one Turing machine that
accepts any input


If A is true, then it correctly claims to be true, and thus is consistent
with being true.

Decidability does not require self-consistency. Not does it require any
particular statement to be true or false.

Likewise, if A is false, then it falsely claims
to be true, and thus is consistent with being false. Thus, the truth or
falsity of A is "undecidable".

On the other hand, if B is true, then it falsely claims to be false, and
thus is inconsistent with being true. Likewise, if B is false, then it
correctly claims to be false, which is inconsistent with being false.
Thus, the truth or falsity of B is also "undecidable".

I think you are confusing "computability" with "decidability".
Certainly it is the case that all uncomputable problems are also
undecidable, but the reverse is not true.


Now, if I understand your argument correctly, you claim, as a human, that
you can step "outside the box" and assign the proper truth value to each
of the above statements. If, so, please show us how and why your
assignments are the correct ones.

As a human being I can step out side the box and say that statement B
is inconsistent.
Cheers,
Randy Hyde

.

Relevant Pages

  • Re: .EXE -> .ASM -> .EXE
    ... :this is a consequence of Godel's incompleteness theorem ... If A is true, then it correctly claims to be true, and thus is consistent ... which is inconsistent with being false. ... the truth or falsity of B is also "undecidable". ...
    (alt.lang.asm)
  • Re: Godel cant tell us what makes a mathematical statement true
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    (sci.logic)
  • Re: Godel cant tell us what makes a mathematical statement true
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    (sci.logic)
  • Re: A Proposed New Principle of Mathematics
    ... > undecidability of a wff in a theory T, which is as strong as PA in ... If this.X is a constant and % is in your language, ... > There exist a model of T where Phi is true, ... How is this going to allow us to "arrive at Incompleteness Theorem"? ...
    (sci.logic)
  • Re: proving that a statement is undecidable
    ... undecidability of his Goedel sentence, ... of axioms if those axioms cannot prove S and cannot prove. ... is a strictly stronger property than the property of undecidability. ... (truth in all models). ...
    (sci.math)