Re: Godel proved maths inconsistent not incompleteness theorem

On May 6, 5:20 pm, herbzet <herb...@xxxxxxxxx> wrote:
Charlie-Boo 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: “ ‘Implies’ and ‘subset’ are the same principle.  There is no
reason to distinguish between a set and a predicate.”

You wrote: “Oh my god.”

Bertrand Russell wrote: “In 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.” - Principles of Mathematics

Yes, this appears in

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

to which I gave the link a few days ago in

   news:48180F8C.364310BD@xxxxxxxxxx

You're welcome.

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?

Russell also writes there:

 "The symbolic affinity of the propositional and the class
  logic is, in fact, something of a snare ..."

Did you read that far?

I read it but I also understood it. Russell wrote earlier in that
same paragraph:

“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.”

So he is saying that an expression (relation) over propositions
E(p,q,r) may be true for all propositions but the same expression over
arbitrary classes a,b,c that is, E(a,b,c) might not. That is because
he is saying that propositions are only zero-place so we have some
expressions that do not hold for classes which are one-place or more.

But we are not talking about taking an expression that is true for all
propositions and claiming that it is true for all classes. We are not
talking about whether an expression is true for all values of its
variables at all. We are not replacing zero-place propositions with
arbitrary one-place or more classes at all.

We are talking about taking an expression that is true of some
particular propositions and replacing each proposition with the class
containing the same elements that hold for that proposition. If
propositions are zero-place as he requires, we would substitute the
empty or universal class and the corresponding E(a,b,c) would in fact
hold.

That is, if A,B,C are each {} or {everything} and A c (B u C) then in
fact A c B or A c C. (If A is empty then A c B regardless of B, and
if A is everything then B is everything or C is everything.)

Once again, he who lives by the sword.

Also from Principles of Mathematics:

 "This seems to show that formal implication involves something over
  and above the relation of implication, and that some additional
  relation must hold where a term can be varied."

Russell thus distinguishes implication from formal implication.

Russell is talking about the difference between,

“Socrates is a man implies Socrates is a mortal.”

and,

“Socrates is a man implies Socrates is a philosopher.”

He asks, they are both true, but what can we replace the word
“Socrates” with?

BR: “We should start with our whole proposition, ‘Socrates is a man
implies Socrates is a mortal.’, and vary Socrates in this proposition
as a whole.”

however,

BR:” “We have ‘Socrates is a man implies Socrates is a philosopher.’
but in this proposition the variability of Socrates is sadly
restricted.”

So he is pointing out that if we have a particular P, Q and k where
P(k)=>Q(k) then sometimes we can generalize that to (all x)P(x)=>Q(x)
and sometimes we can’t.

However, we are not talking about generalizing some P(k)=>Q(k) to (all
x)P(x)=>Q(x) at all, and that distinction is of no consequence. It is
only a philosophical pondering of the difference.

Does CBL distinguish the relation of implication from that
of formal implication?

What is the difference and how is that relevant? (Hint: See above.)

Which of the two does the symbol
"=>" denote in CBL?  How is the other denoted, if at all?

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

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.”

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.

[...]

--
Norm  http://us.metamath.org/email.html(Replyto author
at this URL.  The "from" address in this post is not valid.)

David C. Ullrich- Hide quoted text -

- Show quoted text -

--
hz- Hide quoted text -

- Show quoted text -- Hide quoted text -

- Show quoted text -

.

Relevant Pages