Re: Godel proved maths inconsistent not incompleteness theorem

On Apr 15, 9:32 pm, William Hale <h...@xxxxxxxxxx> wrote:
In article
<69393db2-90a1-41f3-995b-7575c7faf...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,



Charlie-Boo <shymath...@xxxxxxxxx> wrote:
On Apr 15, 4:38 pm, William Hale <h...@xxxxxxxxxx> wrote:
In article
[cut]
For example, "x+0=x" can be proved in PA.
Likewise, "x+0=x" can be proved in ZFC.
But, real analysis cannot be done in PA.

The rules of Arithmetic apply to real numbers. Yes, there is more
than one cardinality of infinite sets and we need a universal set of
at least the cardinality of that of the system we are representing.
But that doesn't say anything about limiting the domain of Arithmetic
functions.

Are you saying that real analysis can be done in PA?

Note the requirements that I listed: "we need a universal set of at
least the cardinality of that of the system we are representing"
There's nothing wrong with the structure of PA as far as real analysis
goes. You just need a universal set of cardinality greater than or
equal to that of the real numbers, and make sure references to (all x)
work for real numbers or explicitly say they are referring to the
natural numbers.

You are trying to make some significance out of the use of ZFC when
there is none. The dozen ZFC axioms have nothing to do with branches
of Mathematics outside of Set Theory. You have the PA axioms for
Arithmetic and the ZF axioms for Set Theory. Then you claim that
covers almost everything else as well.

Yet among that vast subject area of Mathematics you can't find a
single example of a formal ZF proof of anything outside of Arithmetic
or Set Theory for which it was designed. All references have been
bogus.

How would you prove the Pythagorean Theorem using ZFC? That is very
simple standard Mathematics and has at least 20 different proofs. How
would you represent the theorem? How would you prove it, in general
terms?

Let us stop here and examine that question. You make a claim and I
ask for an instance of that claim. So I present one very simple
example from one branch of Mathematics, Geometry. Specifically, how
would you do as you claim is possible: How would you represent the
Pythagorean Theorem? How would the proof go - how would it manage to
prove the formal theorem?

We can see that the ZFC axioms are NOT useful at all for anything
other than proving simple obvious statements about sets.

What nontrivial statements have been formally proven using the ZFC
axioms? In fact, what important questions have been shown to be
independent of and undecidable by ZFC's axioms? Many come to mind.

We know for a fact that there is a great deal of Mathematics that
definitely cannot be determined by ZFC's axioms, for example:

1. The consistency of ZFC.

2. The continuum hypothesis.

3. Martin's axiom together with the negation of the continuum
hypothesis.

4. The axiom V=L (all sets are constructible.)

5. The existence of large cardinal numbers, such as inaccessible
cardinals and Mahlo cardinals.

6. ZFC cannot refute the existence of large cardinals even under the
added hypothesis that ZFC is itself consistent.

7. There are many cardinal invariants of the real line, connected with
measure theory and statements related to the Baire category theorem,
whose exact values are independent of ZFC.

8. There exists a counterexample to Naimark's problem which is
generated by N1 elements. (Akemann and Weaver)

9. Kaplansky's conjecture as to whether there exists a discontinuous
homomorphism from the Banach algebra C(X) (where X is some infinite
compact Hausdorff topological space) into any other Banach algebra.
(Dales and Solovay)

10. The existence of strong versions of Fubini's theorem, where the
function is no longer assumed to be measurable but merely that the two
iterated integrals are well defined and exist.

11. The combinatorial statement <>.

12. The Whitehead problem: Is every abelian group A with Ext1(A, Z) =
0 a free abelian group?" (Shelah)

13. Suslin's problem.

What about these questions of Mathematics - how does ZFC answer these
questions? You said that ZFC could be used to do that, so how would
it?

C-B
.

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