Re: Godel proved maths inconsistent not incompleteness theorem

On Mar 12, 8:33 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 12, 10:06 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:

ZF does nothing useful.

At the very least it provides an axiomatization for a vast amount of
mathematics. If that is not useful to you personally, then so be it.

The only thing ZF does is dinky little stuff we already knew about
sets. Peano's Axioms already existed for arithmetic. Nothing else
uses the ZF axioms. What proof actually uses the ZF axioms outside of
sets and arithmetic? For example, how would you prove the Pythagorean
Theorem using the ZF axioms? Answer: It doesn't do that - it only
does sets and arithmetic.

 It's axioms are not even enough to decide any
useful question about sets.

It depends on what you mean by "useful". In any case, ZF does decide a
vast amount about sets and mathematics.

What does ZF decide that we didn't already know - and is trivial at
that?

 What it does do is to show silly little
statements about sets that we already know are true.  What new fact -
or any fact the least bit subtle - has it shown?

(1) Even if it served only to axiomatize results already known, it
would have value. (2) New set theoretic results are published all the
time. Whether known before or not, one fact is that in Z it's not the
case that for every set S we have the set of tuples (tuples taken as
iterations of ordered pairing) of members of S, but in ZF we do.

The real intent was to say they fixed the Russell Paradox because it
is all formal,

The real intent of what? Of ZF specifically? No, that was not the
intent of formulating ZF.

Why then? The 1st reference I checked is Classic Set Theory,
Goldrei. Page 66: "The theory of sets had alraming and deep
problems. The axioms of set theory are designed to avoid these
problems."

but what they have is no solution as it doesn't provide
the facilities needed in an axiomatization: to be able to decide
questions about Set Theory.  (CBL decides numerous questions about all
sorts of branches of CS.)

It is an incomplete theory, as is any consistent recursively
axiomatized theory in which we can define all the primitive recursive
functions.

But rather than missing queer statements about whether "For All Proofs
If it proves this doesn't halt then do halt" halts or not (which is
what PA is missing), it is missing fundamental questions such as
deciding the Continuum Hypothesis, and we have to spend decades
figuring out it's impossible for ZF to do it - it's just too dinky.
That's the mathematical truth.

What new facts has ZF shown us?
 Rememeber, that's ZF, the 8 or 10
axioms.

All kinds of things are provable in ZF that are not provable in Z. I
just mentioned one earlier in this post.

What has it proven that we didn't already know?

(The real source of the problem is that they are inconsistent about
what a wff can contain.

There's no such inconsistency. The definition of a wff in the language
of ZF is precise.

 "Can a wff contain a reference to something
that is not a set?"

That's not even a coherent question.

Wffs refer to relations (sets.) Can they refer to non-sets there?
Wffs contain operations on relations. Can those operations be done on
nonrelations?

The problem was they figured every wff defined a set based on the wff
being true when a value is substituted for a free variable. Then they
thought of {x|~(x e x)} and there was a wff that was not a set. So
they said, ok, a wff is not always a set.

But there are only aleph-1 wffs and aleph-2 sets of numbers alone, so
there are plent more sets than wffs and we can say that there is in
fact a set for every wff. It is counter-intuitive to say sets do not
include what all wffs define.

In any case, they set out to define what any set can define, when the
real problem was they needed to be exact about terms such as wff.

If you first decide if a wff can make a reference to something that is
not a set, then if yes, then naturally these wffs are not sets, and if
no, then the {x|~(x e x)} example is NOT a set and we still have each
wff defining a set. So we don't really need to worry about what's in
a set as we do in ZF. We just need to be careful about the formal
system - what is the real syntax of a wff? Can it contain references
to nonsets?

C-B

 The Russell Paradox occurs because they are
inconsistent on this question.

You're showing your ignorance and confusion in bright colors now.

Pick an answer (yes or no) and there
is no Paradox if you keep that answer in mind.  They never thought of
it because they didn't even know there was such a thing as a non-set.
In fact, some still say to this day that "Everything is a set."!!  {x|
~(x e x)} is not.

You desparately need instruction in basic mathematical logic,

No, CBL is a lot better.

especially the subjects of improperly referring terms and variable
binding notation.

MoeBlee
.

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