Re: My talk about Godel to the post-grads.

On Jun 27, 10:02 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
MoeBlee wrote:
On Jun 26, 9:03 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
MoeBlee wrote:

 > Anyway, what is an example of a consistent theory that proves its own
 > completeness?

The *only* requirement for the T from Peter's challenge 2) and from *your own*
question is that T is consistent.

No, it is also required that T prove its own completeness.

I did make the correction that T's being required to be complete.
So there really are only 2 requirements: consistent and complete.

No, consistency and proving its own completeness are required. It's
not a matter of the theory being complete, but of the theory ITSELF
proving its completeness. And (as I await clarification on this
point), the theory might not even be complete even though it proves
its own completeness.

"That T prove its own completeness" can't be a requirement here
because we've not agreed on definition of a (general) T's proving
its own completeness.

It IS the requirement since it is the VERY REQUIREMENT that Peter
gave. Peter didn't just ask for a consistent, complete theory; it's
easy to name one of those. Rather, he asked for a consistent theory
that proves its own completeness. As to defininig that, yes, it would
need to be given a precise definition, but that doesn't mean that
Peter simply meant consistent and complete.

And that's my point in answering your original
question! Unlike a consistent or complete system where the definition
is clear cut and indisputable, there's no formal definition for
what is meant by a formal system T proving its own completeness.

I think we could come up with one, with a little bit of work.

My suggestion here, in the 5+ steps, would use first-order proof
(which you'd agree), but doesn't require T to "capture arithmetic",
to present such definition. And I think it's a sound definition.

I don't care about any of your steps since they don't address Peter's
problem. I don't need any of your steps just to show that there is a
consistent and complete theory. There are famous examples already of
consistent and complete theories.

Lacking a common definition, you should look at my suggestion on its own
merit, rather than looking at it from your preferred definition (where
arithmetic is required), which I think is too restricted and wouldn't
reflect the *general sense* of proving T's own completeness

I didn't claim that expressing or capturing arithmetic is a necessary
condition. I only mentioned the lack of expressing or capturing
arithmetic in the sense that if you don't at least do that it is not
seen how you could pull off showing that the theory proves anything
about itself. Simply that I don't see how you'd do it; not that it is
impossible that one would.

Hint: Don't use what you stated below as an answer. Think of what it means,
in meta level, for a general formal system T to prove its own completeness?

Better: Since YOU are making the claim that T proves its own
completeness, YOU should specify what is meant by "Theory T proves the
completeness of theory T".

Sure. T's completeness basically partition the collection of L(T)'s
formulas into maximum 2 equivalent classes: the theorems, and those that
are not.

The issue is not to define 'T is complete', but rather "T proves its
own completeness". Actually, that would be Peter's job. But IF you
propose to have answered Peter's question without him specifying that
definition, then one would think you had yourself some idea of a
definition.

If T is inconsistent, there is only one class: all formulas
are theorems. If T is consistent (but still complete) we need to talk
*only* about one class: the theorem-class. The reason being is without
theorem-hood of a formula F, we can't tell ~F is not a theorem of T!
On the other hand, if F is a theorem of a consistent theory, we know
~F is not a theorem.

Yes, we all know about that. You've done nothing though to fill in a
definition for Peter nor to show an example of such a theory as his
problem ask for.

So then we'd just pick *any* theorem-class formula as a representative of one
completeness-partition, hence a representation of two completeness-partitions,
hence a representation of the entire completeness of T. The problem though
is how do we, *in general*, pick the right formula to represent the entire
theorem class? (The set of theorems due to an axiom, say, A1 doesn't constitute
the entire class!). If T is finitely axiomatizable, the desired formula would
be the one obtained by combining all the axioms by the conjunction "and".

Since in my example of T = { A1 df= Axy(x=y)} there's only one axiom, therefore
A1 is the representative theorem of T's completeness. QED. (And this doesn't
require anything about T's capturing arithmetic).

You've not shown a consistent theory that proves its own completeness.
That's the problem on the table.

He's saying that to
define a function symbol in a language in a way that satisfies
eliminability and non-creativity, the theory must have the appropriate
existence and uniqueness theorem. Your theory could not prove such an
existence and uniqueness theorem. You can't prove the existence of a
function in a theory that asserts that there exists only one object.
And you've not shown how to encode such things as 'sentence',
'theorem', etc. in the theory T.

You omitted responding to the above. And I have no hope that you
understand it, clear and sensible as it is.

As I've explained above, I understood it all right: it's too restricted
(i.e. not generic) a definition!

It's not a definition nor supposed to be one, you hopeless doofus!

MoeBlee

.

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