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From: Charlie-Boo on 1 Jul 2010 00:03
On Jun 30, 6:48 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote:
> On Tue, 29 Jun 2010 18:24:59 -0700 (PDT), Charlie-Boo
>
>
>
>
>
> <shymath...(a)gmail.com> wrote:
> >On Jun 29, 5:25 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:
> >> On Jun 28, 7:04 pm, Charlie-Boo <shymath...(a)gmail.com> wrote:
>
> >> > On Jun 28, 12:44 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:
>
> >> > > On Jun 26, 9:19 pm, Charlie-Boo <shymath...(a)gmail.com> wrote:
>
> >> > > > It would be cool if the following 3 things were equivalent:
>
> >> > > > 1. |- (allX)P(X)
> >> > > > 2. (allX) |- P(X)
> >> > > > 3. ~ |- (existsX)~P(X)
>
> >> > > I can help you here, if you're interested in understanding this.
>
> >> > > (1) is well formed.
>
> >> > > (2) is not well formed as you've given it. The reason is that you've
> >> > > mixed meta-language and object language in an incorrect way.
>
> >> > Did you read the definition of the syntax and semantics in the
> >> > previous post?
>
> >> Sorry, I made the mistake that your first post was intelligible
> >> standalone.
>
> >> > "P(x)" is a wff that is said to be
> >> > provable. So it expresses the proposition that for all values of X,
> >> > the wff P(X) is provable.
>
> >> Then your formulations as given make even LESS sense.
>
> >What is wrong with the proposition that for all values of X the wff
> >P(X) is provable?
>
> Unless there's something you're not telling us about what sort
> of "values" you're considering,

What is a value "external to the language"?

A property can be applied to anything. If the thing being applied to
has no relationship with the property, we haven't established that
that thing has that property and never will, so we say it is FALSE as
it does not "have" that property. Any counterexample?

> the problem is it simply doesn't
> make any sense. If X is an expression in the formal language
> in question then P(X) is reasonable informal notation for the
> result of a certain substitution. But if X is a "value" external
> to the language then there's simply no such thing as P(X).
>
>
>
> >As a 3rd problem, what does it prove to prove that these are not all
> >equivalent?
>
> >C-B
>
> >> I'm out of time for you. I can't do what no one else in these threads
> >> has ever done: get you to understand ANYTHING.
>
> >> MoeBlee- Hide quoted text -
>
> >> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -

From: Charlie-Boo on 1 Jul 2010 00:05
On Jun 30, 5:49 pm, Jan Burse <janbu...(a)fastmail.fm> wrote:
> Charlie-Boo wrote:
> > On Jun 29, 6:36 pm, Jan Burse<janbu...(a)fastmail.fm>  wrote:
> >> Jan Burse schrieb:
> >> Oops, should read:
>
> >>> Herbrand Theorem for a certain class of Formulas A:
>
> >>> |- forall x exists y A(x,y)
> >>> ==>  there are terms t1,..,tn such that
> >>> |- forall x (A(x,t1) v .. v A(x,tn))-
>
> > If A(x,y) is x<y then I would think that the top line is true (do you
> > think so?) but the bottom line certainly is not.  Does this help solve
> > either problem that I posed?  I deal with only a 1-place relation and
> > you deal with a 2-place relation.
>
> If A(x,y) is x<y, then forall x exists y A(x,y) does not
> hold.

Yes it does. y could be the successor of x.

C-B

> The above theorem talks about tautologies and the
> like.
>
> If you have a preorder theory PO or something, then
> maybe something like |- PO -> forall x exists y A(x,y)
> might hold, but this is not the form where the
> theorem applies.
>
> B y e- Hide quoted text -
>
> - Show quoted text -

From: billh04 on 1 Jul 2010 00:41
On Jun 30, 11:03 pm, Charlie-Boo <shymath...(a)gmail.com> wrote:
> On Jun 30, 6:48 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote:
>
>
>
> > On Tue, 29 Jun 2010 18:24:59 -0700 (PDT), Charlie-Boo
>
> > <shymath...(a)gmail.com> wrote:
> > >On Jun 29, 5:25 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:
> > >> On Jun 28, 7:04 pm, Charlie-Boo <shymath...(a)gmail.com> wrote:
>
> > >> > On Jun 28, 12:44 pm, MoeBlee <jazzm...(a)hotmail.com> wrote:
>
> > >> > > On Jun 26, 9:19 pm, Charlie-Boo <shymath...(a)gmail.com> wrote:
>
> > >> > > > It would be cool if the following 3 things were equivalent:
>
> > >> > > > 1. |- (allX)P(X)
> > >> > > > 2. (allX) |- P(X)
> > >> > > > 3. ~ |- (existsX)~P(X)
>
> > >> > > I can help you here, if you're interested in understanding this.
>
> > >> > > (1) is well formed.
>
> > >> > > (2) is not well formed as you've given it. The reason is that you've
> > >> > > mixed meta-language and object language in an incorrect way.
>
> > >> > Did you read the definition of the syntax and semantics in the
> > >> > previous post?
>
> > >> Sorry, I made the mistake that your first post was intelligible
> > >> standalone.
>
> > >> > "P(x)" is a wff that is said to be
> > >> > provable. So it expresses the proposition that for all values of X,
> > >> > the wff P(X) is provable.
>
> > >> Then your formulations as given make even LESS sense.
>
> > >What is wrong with the proposition that for all values of X the wff
> > >P(X) is provable?
>
> > Unless there's something you're not telling us about what sort
> > of "values" you're considering,
>
> What is a value "external to the language"?
>
> A property can be applied to anything.  If the thing being applied to
> has no relationship with the property, we haven't established that
> that thing has that property and never will, so we say it is FALSE as
> it does not "have" that property.  Any counterexample?

I'm not sure what counterexample you are asking about. But, here is my
answer.

Let PA be the system under discussion.
Let P(X) = (some Y)(X = Y).
Then, first statement is: |- (all X) (some Y)(X = Y).
The second statement is: (all X) |- (some Y)(X = Y).
Then, the first statement is true in PA.
Then, the second statement is not true by considering X to something
other than a natural number, since Y must be a natural number.

>
> > the problem is it simply doesn't
> > make any sense. If X is an expression in the formal language
> > in question then P(X) is reasonable informal notation for the
> > result of a certain substitution. But if X is a "value" external
> > to the language then there's simply no such thing as P(X).
>
> > >As a 3rd problem, what does it prove to prove that these are not all
> > >equivalent?
>
> > >C-B
>
> > >> I'm out of time for you. I can't do what no one else in these threads
> > >> has ever done: get you to understand ANYTHING.
>
> > >> MoeBlee- Hide quoted text -
>
> > >> - Show quoted text -- Hide quoted text -
>
> > - Show quoted text -- Hide quoted text -
>
> > - Show quoted text -

From: Jan Burse on 1 Jul 2010 04:38
Charlie-Boo schrieb:
>> If A(x,y) is x<y, then forall x exists y A(x,y) does not
>> hold.
>
> Yes it does. y could be the successor of x.
>

But this means you are not working in the empty theory.
You are working in peano or so. But the Herbrand Theorem
is formulate with nothing in front of the turnstile (|-).
So we do not assume peano, and thus cannot follow
forall x exists y A(x,y).


From: Charlie-Boo on 1 Jul 2010 09:40
On Jul 1, 5:56 am, Frederick Williams <frederick.willia...(a)tesco.net>
wrote:
> Charlie-Boo wrote:
> > > >> > > > 2. (allX) |- P(X)
>
> > What is a value "external to the language"?
>
> There are symbols (of the language) and there are things (not of the
> language) symbolized.  Logic text books are wont to point out the
> difference between London (a place) and 'London' the name of the place.

They are talking about the differences in the representations not so
much the values. What they are really talking about is the difference
between a program and its output. "the bible" is a program, and the
string between the previous two quotes is the output.

But because they have never formalized even the simple primitive
concept of the relationship between a program and its outputs, they
fumble with a new word for it and never see that relationship in
general (and I had to extend logic into CBL to express these
relationships between sets of differing cardinality.)

Recursion Theory deals with the cases in which they are the same. We
start with a program that outputs a copy of itself - the program
equals the output.

But I see what you're saying and it's helpful, thanks. My immediate
thought at the original question was that he's just harping over the
fact that I didn't give the universal set. I thought of a few
possibilities, e.g. it doesn't change the answer to my original
question of proving that these 3 wffs cannot be equivalent, or giving
a formal definition of the syntax and semantics i.e. definitions of
the wffs that are now allowed. (I gave a short informal one.) But a
program accepts a set and programs are different from sets. If/when I
get back to analyzing the question posed, I will remember that we are
talking about my friend yes(I) which is the set of values that program
I (for input) accepts, expressed in CBL. (What is the expression for
that in conventional math? Ha!)

In the meantime, if I just said that the universal set is N (which was
my intention, but thanks for pointing out that there are other
possibilities of interest), would everybody be happy? (And then start
using CBL instead of that dinky little ZFC nonsense?)

I think that part of the problem is those who can understand only what
was drilled into them in school (if that) and can only prove what they
were shown how to prove. Going beyond that is a long, tedious
process. Witness the lack of any new proofs for existing significant
theorems on these pages except from you-know-who.

This is similar to those famous quotes about people being afraid of
new ideas, those who love ideas vs. those who fear them, etc. In this
case they are unable to fathom them until . . . well, the quote about
science advancing "one funeral at a time" comes to mind.

But that adherence to the past may have some applicability to the
future.

C-B

> --
> I can't go on, I'll go on.

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