Re: Godel proved maths inconsistent not incompleteness theorem

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.

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". 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 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.

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
.

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