Re: Godel proved maths inconsistent not incompleteness theorem

On Apr 18, 4:40 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
Charlie-Boo says...

Why is it *bad* that the ZF proof of
the Pythagorean theorem is only a few lines long?

It isn't.  If you mean why isn't it ok to define what you're trying to
prove and then you have a one line proof, I assume you know the answer
to that.

No, I have no idea what you are going on about,
and I don't think you really know what you are
asking for, either.

And I think you're a horse's ass for making personal attacks during
what should be an intelligent discussion of principles of Mathematics.

Would it be ok to define the sum of the angles of a triangle to be 180
degrees? Or to define the distance between two points to be the
square root of the sum of the squares of the difference between the
two coordinates? That is the distinction I am talking about - use vs.
misuse of definitions.

You asked whether ZF proves the Pythagorean
theorem (when point, line, etc. are interpreted as
particular types of sets). The answer is: yes, it's
provable in just a few lines. Every standard theorem
about Euclidean geometry is provable.

Ok, here's a simple one: For any finite set of points in a plane there
is a line that contains all of them or exactly two of them.

I think that a healthier attitude than being so cavalier as to declare
all of Mathematics to be provable by ZF is to:

1. Ask what can be proven in ZF.

2. Drop the idea that ZF’s 10 axioms have anything to do with it
(outside of Set Theory). It’s all from Peano’s Axioms and the use of
sets to represent real numbers.

3. Recognize that Peano’s Axioms amount to the 3 relations of
addition, multiplication and TRUE (the universal set) being
representable (without reference to the actual axioms.)

Has # 3 ever been published or discussed (outside of my mentioning it
occasionally)? I have yet to sit down and try to replace PA based
proofs with this way of expressing Peano’s Axioms. I would be
interested in any effort to do that. I think a good first step is to
pick a handful of simple theorems with simple proofs to try.

One consequence of # 3 is that any true sentence of the form A+B=C or
A*B=C is provable. This is also true of certain wffs that use only
these relations, but I can’t say offhand which ones. CBL has rules
for making conclusions concerning r.e. sets (see ARXIV paper and my
example proofs), so if it is shown to be r.e. then this is true. (I
am assuming that the Godel Sentence doesn't appear in there somewhere
to interfere.)

In general that means no universal quantifiers, but I don’t know about
the case where you use only relations ADD and MUL (addition and
multiplication.) Is there a theorem for that? That is, which
sentence that use only addition and multiplication are provable when
true?

C-B

People gotta be above immature behavior such as attacking anyone who
disagrees with conventional wisdom. (Yes, I realize it has been going
on since the beginning of time - but I would hope that supposed
scientists would be better than such baby-***.)

--
Daryl McCullough
Ithaca, NY
.

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