Re: Godel proved maths inconsistent not incompleteness theorem

In article
<aa0d86f1-0891-4e52-a44e-f332a9456de0@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Charlie-Boo <shymathguy@xxxxxxxxx> wrote:

On Mar 27, 11:28 pm, William Hale <h...@xxxxxxxxxx> wrote:
In article
<aaf23217-5969-4ae5-bfd6-454d3391a...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,





 Charlie-Boo <shymath...@xxxxxxxxx> wrote:
On Mar 27, 3:04 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 26, 8:42 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

On Mar 25, 2:38 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 24, 10:49 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

On Mar 24, 3:56 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 24, 10:10 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

Ok, help me learn.  What are the formal representations of
the
theorems and where (page of last line of proof) are they
proven?

To put in the strictly formal language requires some tedious
work.
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
Then the proof of the theorem
in ZFC is not difficult.

That is the same mistake that is made routinely around here.

You've shown no mistakes.

 First:
Does this undifficult proof use only ZFC's axioms or do we make
up
suitable axioms ourself?

I've told you about a hundred times already. Please stop asking
over
and over questions that have already been answered. The proofs use
only first order logic applied to axioms of ZFC,

Ok, you say that only the ZFC axioms are used.  Then you are going to
have a really hard time because:

1. You now have to do under more restrictive conditions what hasn't
been done in almost 2,000 years: Axiomatize a great deal of
Mathematics - but using only ZFC's axioms.

No, ZFC does axiomatize a great deal of mathematics.

It's just silly to think that.  Each branch of Mathematics has its own
axioms.  In a way, that's what defines a branch of Mathematics: its
unique set of axioms.

2. Furthermore, the axioms which were not attainable in the past will
now have to be generated automatically from ZFC's dinky little set
that was designed only for Arithmetic and Set Theory.

I don't know what you mean by "generated automatically".

Only ZFC's axioms are used and they already exist.

And the
axioms of ZFC were early recognized as providing for not just
arithmetic but for analysis. Anyway, ZFC either proofs the axioms of
certain systems or expresses those axioms as part of a definition of a
certain kind of structure. For example, where first order group theory
has certain axioms, ZFC has those axioms incorporated into the
defintion of 'group'.

So now you're saying the opposite: we add axioms, in the form of
"definitions".  But a definition means a single symbol with an
expression equivalent to it.  You need to prove yout point

What is the "point" you are referring to?

and give
that axiomatization.

What is the "axiomatization" that you are referring to?





So where group theory proves, say, 'Ax - -x =
x' (where '-' had been defined), in ZFC we prove 'If <S +> is group,
then AxeS - -x = x' (where we would have also defined '-' for groups).

Are you up to it?

You don't seem to be up to opening page one in a textbook where it all
starts.

Yet another bogus book for my collection?  Are you collecting
royalties or something?  (That would explain why you refuse to give
even 1% of what you claim I will find if I buy a copy.)

C-B

Again, quoted by you, but not responded to by you:

which are in the
primitive language, plus defintions that extend the primitive
language
by abbreviatory symbols. Once more and now please go SEE FOR
YOURSELF
by getting a textbook or few and stop asking like a silly child:

We have classical first order logic (which includes identity theory
with a fixed semantics) in the language also of the primitive
relation
symbol 'e'. Then we have the set of axioms of ZFC, all in that
language, with specific axioms and one axiom schema. Morevover, we
may, as we feel suited, abbreviate certain expressions through the
use
of defined formulas, as this may be accomplished formally by the
extending the language by use of definitional axioms (which only
provide abbreviatory power and do not add to the expressive or
deductive power of the language and theory). Ordinarily, after
getting
warmed up with the first few proofs of theorems, we begin to use
informal versions that we can, if we choose to endure the tedium,
convert to pure formulas and purely formal proofs as described
above
in this paragraph.

Now why don't you go learn it like everyone else does who is
informed
on this subject rather than to continually wallow in your
ignorance.

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

An axiomatization of First Order Group Theory.

I don't know if you intended this, but I don't think First Order Group
Theory can be developed in ZFC set theory.

But, I don't think First Order Group Theory is part of what we are
calling standard mathematics. I am a group theorist and this is the
first time I heard of First Order Group Theory.

Even better, prove the Pythagorean Theorem using ZFC. Last I heard
there were at least 2 dozen ways to prove it using very ordinary
Mathematics.

I gave a sketch of this months ago. I forget what the end result was.
I don't think you responded to it.
.

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