Re: Godel proved maths inconsistent not incompleteness theorem

On Mar 28, 3:40 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

How can I be quoting from these books without opening them?

You need to open to page one and study to UNDERSTAND. Just quoting
passages from the middles of various texts is not a basis for
understanding.

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).

What??  There is one passing reference to ZFC in Enderton's book in
section 2.6 and no proofs in ZFC at all are given, much less formal
ones, much less theorems concerning anything outside of Set Theory and
Arithmetic.

I said that Enderton's logic book provides for the PURPOSE of studying
mathematics as formalized in ZFC. I didn't say Enderton's logic book
is ITSELF in formalized ZFC. My reason for recommending that you study
Enderton's logic book is so that you can have an understanding of
mathematical logic. That understanding would then inform you as to
what a formal theory IS and also to see how informal arguments may be
formalized.

Anyway, what you find in Enderton's logic book is formalizable in
ZFC.

Suppes discusses ZFC and its redundancies, and also
provides no formal proofs in ZFC.  The only proofs using ZFC seem to
be proofs that ZFC is redundant.

I said you should FIRST learn the predicate calculus. Then you can see
how the arguments in Suppes's book may be formalized (though Suppes
DOES give at least an outline on that subject). As to redundancy, the
pairing axiom and schema of separation are indpendent from the other
axioms of Z set theory but not from the other axioms of ZF, so for an
independent axiomatization of ZF, just delete the pairing axiom and
the schema of separation.

Now why don't you and Peter Smith stop wasting people's time with BS
references and either spell out at least a fraction of a formal proof
in ZFC of something outside of Set Theory or Arithmetic, or admit that
you have none?  Seems to me that would be the honest approach.

The honest and WORTHWHILE thing for you to do is read textbooks (I
gave you an easy four book starter course) that would inform you of
the subject. Then, once you understand the predicate calculus and what
a formal proof IS, then you may tell us the very first proof in set
theory you don't see how to formalize. We'll help you from there, our
time and patience permitting.

This is just one more example why people need to present the results
claimed here, instead of just giving the title of a book or article.
Simply giving titles doesn't prove anything and lets dishonest people
divert the attention of the person making a point away from the
subject matter at hand.

Just giving a title of a book is not proof. But giving titles of such
good books as I mentioned provide you with a starting point to learn
what formal proof IS and to see how to formalize mathematical
arguments yourself. THEN, once you've done that, go ahead and let us
know of the VERY FIRST proof in Z set theory that you can't see how to
formalize.

If someone gives a bogus reference like you, Peter Smith and others
have,

We've not given you an "bogus" references.

should people continue to track down these references, examine
them, and spell out what's in the entire book or article to prove it
doesn't have what is claimed?  The answer is No, the burden of proof
is on the person making the claim.

The burden is on YOU to learn even what a formal proof IS. And such
books as I mentioned show you how to formalize mathematical proofs.

MoeBlee

.

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