Re: Godel proved maths inconsistent not incompleteness theorem

On Mar 28, 3:15 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

No, the claim was that FORMAL proofs of the cited theorems based on
ZFC's axioms have been published.

That wasn't MY claim. I've said, probably over twenty times now, that
what we typically find are informal arguments from which we, as
informed in mathematics and set theory, can see to provide formal
proof were we to take the time and tediousness to put in purely formal
terms.

As far as believing that it could be formalized in ZFC, you have to
examine the ZFC axioms and see what they say about the subject matter
once you have drawn a correspondence between its objects and ZFC
wffs.

No, all I have to do is follow the train of proofs from the ZFC
axioms.

 That is the problem.  You will need axioms that codify the
properties of that system, and there is no prima facie evidence that
the relationships expressed by the ZFC axioms will give you the facts
needed in the other system.

All I have to do is follow each proof - one to another - until I get
to the proof of whatever theorem under discussion.

As a trivial example, there may be more axioms needed in that system
than the 8-10 axioms provided by ZFC.

It's not NUMBER of axioms, but rather the EXPRESSIVENESS of the
language and the deductive STRENGTH of the axioms.

 Even if the ZFC axioms stated
useful information about that subject matter, there are simply not
enough of them.

Again, it's not a matter of NUMBER. (Aside from the technicality that
actually ZFC has a denumerable number of axioms.) And you merely
ASSERT that ZFC is not enough. Meanwhile, though it can't convince
you, I HAVE personally verified trains of proofs from the ZFC axioms
into areas of mathematics such as analysis, abstract algebra,
topology, and graph theory. That's a lot of pages in a lot of books. I
can't do it for you; if you want to see that it is done, you have to
read the books yourself.

 Reality is, though, there is no reason to believe
that the manipulations expressed by the ZFC axioms are the least bit
helpful in the other system, other than e.g. Induction being useful
when dealing with an enumerable set.

No, I LOOK AT the reality, as I look at the ACTUAL trains of proofs.
You just put your hands over your eyes by refusing even to look at
page one.

That is why nobody can give any examples of a ZFC proof of anything
outside of sets and arithmetic, for which the ZFC axioms were
designed.

We've given examples.

ZFC is no magic bullet.  It is ridiculous to think that almost all of
Mathematics follows the same pattern as that dictated by the ZFC
axioms.

But we've seen almost all of said mathematics in a train of proofs
from the ZFC axioms. Of course, we can't convince you if you refuse to
look at such a train of proofs yourself.

And AGAIN, for about the fourth time, quoted by you but not responded
to by you:

 a mixture of formalisms and natural
language. One can then see that the write up is convincing that a
formal proof exists.

Anyway, you quoted all of this without any response from you:

That's tedious work. The work is
in unpacking definition after definition to get back to the primitive
language. If I don't have to unpack the definitions then stating the
theorem is as simple as this:

Ab(Bb -> Efh(Ff & Ihbf))

Where 'B' is for 'is a Boolean algebra; 'F' is for 'is a field of
sets'; and 'I' is for 'is an isomorphism from to'.

So, then I'd give the formulations (the formal defintions) for 'B',
'F' and 'I'. Then for all defined terms in those definitions, etc.,
unitil wereach a formula in the primitive language of set theory. And
that is a process I'm not going to do for you in just a post, since
it's one that you would carry out for yourself as you study the
material, were you interested, for whatever reason, in the seeing the
full primitive formula.

Meanwhile, I wrote the following, but your lack of comment leaves me
uninformed whether you understand any of it or whether it's worth the
effort providing you with such explanations::

Ordinarily, we use informal renderings that we can see to be
formalizable. That we can see that our renderings are formalizable
comes from our experience first working in the purely formal language
for the beginning theorems and then gradually becoming more informal
as we prove more theorems, as by this time we can see which informal
expressions adequately capture purely formal expressions.

Now, if you have a particular theorem (such as one that has been
mentioned) that you'd like to see how to prove and that it can be
formalized in ZFC, then just get any textbook that proves the theorem.
Then go back to a set theory textbook to see the previous theorems and
definitions leading up, then take that formalization and apply it to
the subject matter of the textbook (or to any textbooks leading up to
your chosen textbook if you've not chosen a beginning textbook in the
subject).

I already gave you an example: A set is recursive iff both the set and
its complement are recursively enumerable. So, to formalize, you first
need to define 'complement', 'recursive' and 'recursively enumerable'
in the language for ZFC extended by definitions. If you read a set
theory textbook, you'll get the supporting theorems and definitions,
including 'relative complement', then you can easily (though perhaps
tediously) put 'recursive' and 'recursively enumerable' in the
language of ZFC extended by definitions. Then the proof of the theorem
in ZFC is not difficult.

Aside from computerized systems such as Norm Megill's, probably the
only managable way to see that a mathematical statement is formally
provable in ZFC is to - GUESS WHAT? -  STUDY set theory and then
mathematics in light of set theory. We do that work, but we can't do
it FOR you. If you want a course of study that will provide you with
this common mathematical understanding then I suggest (in this order):

'Logic: Techniques of Formal Reasoning' - Kalish, Montague and Mar
(for basic skill in the predicate calculus)

'Elements Of Set Theory' - Enderton, along with 'Axiomatic Set Theory'
- Suppes. (For the purpose of studying mathematics as formalized in
ZFC, just the first half or so of those books would be okay - covering
through the construction of the real numbers.)

'A Mathematical Introduction To Logic' - Enderton (For basics in
mathematical logic aside from just basic skill in the predicate
caclulus; and for the purpose of studying mathematics as formalized in
ZFC, the first few chapters of this book would be okay).

Then, with that very basic background, choose just about any
introductory textbook in abstract algebra, topology, analysis, graph
theory, recursion theory, model theory, etc. and you can see how it
can be formalized in ZFC.

MoeBlee
.

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