Re: Godel proved maths inconsistent not incompleteness theorem

On May 6, 5:43 pm, William Hale <h...@xxxxxxxxxx> wrote:
In article
<c5b1f380-ee70-4a05-a3d3-1556de3e0...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,





 Charlie-Boo <shymath...@xxxxxxxxx> wrote:
On Apr 28, 7:56 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Sun, 27 Apr 2008 11:23:52 -0700 (PDT), Charlie-Boo

<shymath...@xxxxxxxxx> wrote:
[...]

If you want to know the truth of the matter, any programmed
implementation of CBL will use only one symbol for both "implies" and
"subset", as there is no ambiguity and they are the same principle.
In fact, there is no reason to distingush between a set and a
predicate.

Oh my god.

I wrote: ³ ŒImplies¹ and Œsubset¹ are the same principle.  There is no
reason to distinguish between a set and a predicate.²

You wrote: ³Oh my god.²

Bertrand Russell wrote: ³In any symbolic expression, the letters may
be interpreted as classes or as propositions, and the relation of
inclusion in the one case may be replaced by that of formal
implication in the other.² - Principles of Mathematics

A url for the full passage of that quote is at:

http://fair-use.org/bertrand-russell/the-principles-of-mathematics/s13

I think you are misrepresenting what Bertrand Russell is trying to say
by taking that statement of his out of context.

What is the context of that statement that needs to be restored?

Russell’s subsequent reservations are only talking about taking a true
expression (all x)P(x) where x ranges over zero-place propositions,
and replacing it with an expression (all x)P’(x) where x ranges over
one-place or more classes. P(x) can be true for all x because each x
is TRUE or FALSE, but P’(x) is not true for all x because each x can
be a class in general.

This has nothing to do with our replacing a one-place or more
proposition with the equivalent class (whose elements are those for
which the proposition holds.) Russell writes,

“If p, q, r are propositions, and p implies q or r, then p implies q
or p implies r.” This proposition is true; but its correlative is
false, namely: “If a, b, c are classes, and a is contained in b or c,
then a is contained in b or a is contained in c.”

The above is true only because his propositions are zero-place but his
classes are one-place or more. I explain this and more in the next
post.

The correspondence remains until someone points out a statement that
is true in one form (proposition/implication vs. set/subset) and not
the other. (Norm tried but failed.)

C-B

Were you as alarmed when Bertrand Russell said it as you were when I
said it?  Or is it merely a blind condemnation if I say it, and blind
praise if a professor says it, with no mathematical significance?

What you are describing is the unfortunate fact that Logic and Set
Theory are distinct branches of Mathematics, when in fact they are the
same principles with only slightly different "interpretations".  There
is a parallel between the two - the theorems are generally in one-to-
one correspondence e.g Demorgan's Theorem, Double Negation etc. apply
to both sets and propositions.

I am not about to make the mistake of using different symbols when
saying ~(P^Q) is equivalent to (~Pv~Q) when P and Q represent sets vs
propositions.  I don't even distinguiosh between the two.

Yes, I don't do it the "standard" way.  The standard way is wrong - it
is poorly designed.  Saying that something is wrong if it isn't the
same way that it has been done in the past is not a sound principle of
system design.  It is an impediment to making improvements.

C-B

[...]

 >     QED                    Therefore the system is not both consistent
 > and complete.

By the way, you haven't defined "system", so the claim of your theorem
is at best ambiguous, because it isn't true for all systems.

As in any discussion of Metamathematics, it holds for any system that
meets the axioms.

 Classical
propositional calculus is a system, and it is both consistent and
complete, contradicting your conclusion.

Propositional Calculus doesn't satisfy the axioms.

[...]

 > Einstein said,
 >
 > "The skeptic will say, 'It may well be true that this system of
 > equations is reasonable from a logical standpoint, but this does not
 > prove that it corresponds to nature.'  You are right, dear skeptic.
 > Experience alone can decide on truth.²

Einstein's quote is about physics, not mathematics.

He doesn't say anything about physics.  Do you think physics is
peculiar in this regard?

I don't know the
complete context of this quote, but I would guess

Perhaps it would be better to use principles of logic rather than
guessing.  Can you see why it is important that a "system of
equations" correspond to reality?

C-B

it means that if your
starting physical assumptions about nature are wrong, no amount of fancy
math will make up for it.  I doubt that he had any reservations about
the math itself.

[...]

--
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at this URL.  The "from" address in this post is not valid.)

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