Re: Godel proved maths inconsistent not incompleteness theorem

On Mar 26, 5:16 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Tue, 25 Mar 2008 13:07:37 -0700 (PDT), Marshall
<marshall.spi...@xxxxxxxxx> said:





On Mar 24, 6:25 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:

[...] it is utterly untethered to the propositions you want them to
express.  How does this tethering happen in real mathematics?  By
building an elaborate infrastructure, beginning, in the case at hand,
with a definition of a Turing machine, a definition of halting, and a
definition of solvability.  These do not exist in CBL, hence the strings
in your "proof" don't express anything approaching what you say --
except, of course, by the Magick of your intentions.

I see the point you are making here. But I have a question about it.
It's not a "devil's advocate" kind of question where I have some
agenda behind it; I just want to understand the issue better.

It seems to me that when we have an abstract algebra such as a group,
we have something that is also in and of itself "untethered" to
anything.

Well, "tethering", as I was using it, is a relation between a formal
system and mathematical reality -- for example, the relation between the
axioms for group theory and the class of groups that those axioms pick
out.

And that's exactly what CBL is doing. As I have said repeatedly, the
theorems apply to any system for which the CBL axioms (of your choice)
hold.

Of course, we know that the abstraction "group" came about as a result
of a deliberate process of abstraction from various mathematical
models. But that's actually what Charlie is claiming he did as well:
studied computability from a highly abstract level and axiomatized it.

The way I actually developed the Rules for the Theory of Computation
was to start with the problem of Program Synthesis. As I detailed
recently, I took a dozen simple computer programs that were well
understood. Over a period of 21 months I represented these programs
and the functions they computed in numerous ways. When I represented
them as a tree-structured flowchart - Eureka! I was able to decompose
them all into single programming constructs e.g. IF X<Y . . . which
are combined by 8 Rules of Inference. The Axioms were simply these
primitive constructs provided by the Programming Language.

I then tackled the Theory of Computation and immediately discovered
that the same Rules of Inference apply. After all, the Theory of
Computation is all about what computer progrmas (Turing Machines)
exist. Instead of primitives of a Programming Language, the axioms
concerned the relationship between a program, its input and output.
They are all along the lines of ideas concerning Mathematical Logic
developed over the last 100 years or so.

YIT(I,J,K)* The relation "Program I with input J halts yes at
iteration K." is recursive. (Kleene T function)

TRUE(x) The universal set is recursively enumerable. (Peano's
axioms.)

[Note that Peano's Axioms may be extracted from a program that lists
the natural numbers. I call this Axioms from Programs.]

This gives you theorems of the form "Set . . . is recursive (or is
r.e.)." For theorems of the form "Set . . . is not recursive (or
r.e.)" I added one Axiom: -~YES(x,x) The set of Turing Machines that
do not halt yes on themselves is not recursively enumerable. This is
really reduction, but to this set being r.e. rather than to the
halting set being recursive. This is from a simple diagonalization
over the set of Turing Mahines that don't halt yes on themselves.

Except he didn't.  The undefined primitives are far too underspecified
to characterize anything even as general as a group, let alone far more
structured and complex notions as those of computability theory.

You only need to codify the properties of the sets involved, not their
construction. You are at too low a level of abstraction. CBL applies
to any system for which the axioms hold.

Now, in the case of groups, we can easily see that, say, the integers
with addition qualifies. That's the tethering.

I don't see what's tethered to it.  What formal system do you have in
mind?

In contrast, we don't have any evidence that CBL is tethered in the
same way to computation at some high level, but I don't see that we
have any evidence that it *isn't* either.

Well, that observation there is itself pretty good evidence.  There is
nothing but his say-so that connects the expressions of his theory and
the mathematical realities he wishes them to be about.

The axioms do that.

C-B

Clearly with Charlie there are some communication challenges. He seems
entirely unconcerned with learning standard terminology and notation
and conforming to it. This is a significant problem; significant
enough it seems to me to sink any chance at recognition if Charlie
does in fact actually have something.

It's evident after even a cursory examination of CBL that he doesn't
"have something".

But other than *behaving* like a crank, with all the raised-by-wolves
manners, the I'm-right-and- everyone-should-change-to-my-way attitude,
etc., I don't see him reaching any obviously wrong conclusions. Am I
missing something?

The obviously wrong conclusions are his claims about what he has proved,
e.g., the unsolvability of the halting problem.  He may have offered
some simple, formally valid proofs, but they are completely unconnected
(save through his intentions) to any actual mathematical theorems.- Hide quoted text -

- Show quoted text -

.

Relevant Pages

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  • Re: Godel proved maths inconsistent not incompleteness theorem
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