Re: Godel proved maths inconsistent not incompleteness theorem

On Mon, 24 Mar 2008 22:42:59 -0700 (PDT), Charlie-Boo
<shymathguy@xxxxxxxxx> said:
On Mar 24, 1:38 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Mon, 24 Mar 2008 09:31:45 -0700 (PDT), Charlie-Boo
<shymath...@xxxxxxxxx> said:

On Mar 17, 11:38 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
On 2008-03-17, in sci.logic, Charlie-Boo wrote:

(As far "standard terminology" goes, the words are something
poorly chosen and I like to avid that. E.g. a decidable wff has
nothing (directly) to do with a decidable system - numerous
authors have apologized for the confusion. In my personal
research, I use more consistent terminology. (A "decidable"
system is one in which all sentences are decidable.) Again, it
just works better.

What do you call, in your personal research, a decidable system in
the usual sense?

PR(I)*

What is its definition? What is its semantics? Why can it not express
"I makes a fine cup of hot chocolate"?

You gotta read the ARXIV paper, as well as my dozen detailed
explanations. (I now realize that many people will never understand
it until I post it to my home page with a zillion pages of detailed
examples and explanations, including clarifications based on readers'
feedback.)

No, people will never understand it until it has genuine, rigorous
mathematical content.

Now, I can say that P is a variable that ranges over relations,
right? That is a very common thing to do in Mathematics - a variable
with a range. I don't have to define what a relation is or what a
variable is or give a formal system that explains it, do I?

Well, it depends on what you are doing with it. Typically, no, one
doesn't have to define what a relation is, unless one uses the term in a
way that cannot take them to be sets of ordered pairs. And, typically,
one doesn't have to say what a variable is. It's not clear at all that
you are using them in a standard way in CBL. Your notation is quirky
and nonstandard, so one cannot be at all sure you have the usual notions
of relation and variable in mind.

After all, we all know what a relation is, what a variable is, and a
variable can range over relations. When someone does this in a
published article, you don't write the editor of the journal and
complain that the author didn't include a whole bunch of background
material, right? So let's try to be reasonable and consistent.

Say that the degree of P is 1, so it is just a set. Then the
following expressions have the following meanings:

P(x) P is recursively enumerable.
P(I)* P is recursive.

And thus we have magickal thinking. I will grant you that, other things
being equal, you don't owe us a definition of "relation" and "variable".
But "recursive" is a whole other kettle of fish -- at least in the sort
of foundational system CLB purports to be. No, you need to define that
notion in terms of initial functions are the usual operations, or Turing
Machines, or a Turing-complete programming language, or what have you.

When we are proving results concerning a Logic, we let PR = the set of
theorems

Theorems of *what system*?

and TW = the set of true sentences.

Sentences in *what language*? True in *what model*?

And we don't give "the" formal system because it applies to all formal
systems that meet the axioms given in CBL that refer to PR and TW. We
simply give axioms.

Sorry, that's just not going to wash. The sort of thing you are talking
about can indeed be done in a general way, but you need to put far
greater constraints constraints on what counts as a formal system, what
it for a sentence of such a system to be a theorem, what it is for a
sentence is to be true in an interpretation of the language of such as
system, and so on. Otherwise nothing tethers "PR" and "TW" to the
theorems or truths of any given system besides your say so.

(I can see at this point that it is used a little differently than the
typical axiomatic system: Typically we have a single Logic and axioms
about that Logic. Here we are saying that we have any Logic that
satisfies the axioms.)

But you need to say what a Logic is for that claim to have any purchase.

Now, we can give axioms for PA or we can just give axioms that hold
for a typical Logic. (This is VERY common.) We can say things like:

PR => TW The Logic is sound.

What ensures that PR doesn't refer to the set of true sentences of a
system (in what interpretation?) and TW the set of theorems? Answer:
NOTHING. Not until you build your system from the ground up a produce
rigorous definitions of truth (in an interpretation) and theoremhood in
a system.


PR(x) The set of theorems is recursively enumerable.
PR(I)* The set of theorems is recursive.

I have no idea what "x" and "I" are doing here, but what's to prevent:

TW(I)* The set of truths is recursive.

Answer: NOTHING. Because you haven't defined "true" and "recursive".

-TW(x) The set of true sentences is not recursively enumerable.

What if the system in question is Presburger Arithmetic -- where the set
of true sentences *is* r.e.?

Again, we might be talking about PA and give correct axioms for PA, or
we can simply say that any Logic for which our axioms hold has the
properties we are about to prove from these axioms.

Again: Nothing tethers the strings above to your intended meanings.
There is no mathematics there. It is just magickal thinking.

Now define variable YES by YES(x,y) iff Turing Machine x halts yes on
input y. Again, it is just a variable that is given a fixed value for
this discussion. (And for heavens sake we don't need to copy Turing's
definition of a Turing Machine here.)

Oh, I beg to differ. That is *exactly* what you need to do. And in
particular you need to use that definition to provide rigorous
definitions of "recursive" and "r.e.". Not to mention the fact that you
need to define what a Turing machine so you can enumerate them and
thereby make sense of the notion "Turing Machine x".

In general, if P is a 1-place relation and Q is a 2-place relation,
then P/Q means there is an M such that P(a) <=> Q(M,a) [for all a].

So, for example, let P be the set of primes and let Q be the set {<0,p>
| p in P}. So we have P/Q. Ok.

We can say that "M characterizes P in (base) Q."

So the number 0 characterizes the set of primes in base, uh,
set-of-pairs-<0,p>-for-any-prime-p? Well, ok.

Then the expression P/YES means that P is recursively enumerable.

Yeah, I can see that that is what that would mean if you worked out the
details.

And if Q is other relations besides YES, P/Q means that P is
representable or expressible, as you can perhaps see.

Perhaps I could, if you actually defined what it means to be
representable -- in what language? Relative to what interpretation?
The natural numbers? Where do you say?

P(x) is actually an abbreviation for P/YES.

(I'd be glad to discuss in detail the two approaches. But formally
expressing a property instead of giving a "kludge" definition of that
one case is not only easier to work with, the special particular
definitions actually interfere with proofs that use the general,
formal approach.)

Except CBL doesn't "formally express" any properties at all.

CBL is a formalization of Metamathematical Logic.

Oh, but it isn't. It really isn't. It's nowhere close to a
formalization of metamathematics. It is *at best* a wan gesture in the
direction of such a formalization, though I think even that is an
generous overstatement. If you want a formalization of metamathematics,
you need to formalize, say, Mendelson; or Enderton; or Kleene; more
generally, books contain the actual mathematics of metamathematics.
Though it would be frankly crazy to do so, unless you have, say, a
kickass automated theorem prover and you want to see how well it does
with metatheory.

PA formalizes part of Mathematics and CBL formalizes Metamathematics.

Sorry, it does no such thing.

.

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