Prev: Cantor's First Proof
Next: math symbols in unicode collection
From: Charlie-Boo on 1 Jul 2010 09:43
On Jul 1, 12:06 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Charlie-Boo <shymath...(a)gmail.com> writes:
> > What is a value "external to the language"?
>
> The expression (all X) |- P(X) makes sense only when we're discussing
> provability of formulas in a language that includes a name for each
> possible value of X.

Does (existsX)P(f(X)) make sense if f is not total?

C-B

> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Charlie-Boo on 1 Jul 2010 09:51
On Jul 1, 12:41 am, billh04 <h...(a)tulane.edu> wrote:
> 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.

A property and a thing to which the property neither appies nor
doesn't apply. Oops, I may have just thought of one.

> 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.

Do you think that the quantifiers refer to values outside of the
universal set? How do you define truth when the expression has a
quantifier?

(Hmmm . . . If ZFC can't have a universal set, how can it have even
quantifiers??)

C-B

>
>
>
>
> > > 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 -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -

From: Charlie-Boo on 1 Jul 2010 09:55
On Jul 1, 9:49 am, Frederick Williams <frederick.willia...(a)tesco.net>
wrote:
> Charlie-Boo wrote:
>
> > In the meantime, if I just said that the universal set is N (which was
> > my intention, ...
>
> Look,
>
>  (allX) |- P(X)
>
> is hard to understand _whatever_ the domain of quantification (is that
> what you mean by 'universal set'?).  Does it mean something like
>
>  whatever term is substituted for the free variable X in P,
>  the result is provable
>
> or what?

I thought I said it, but does this work: Read it left-to-right,
reading substrings that form familiar mathematical concepts. In this
case we have "(allX)" which is . . . etc.

Yes, of course it means that. How would you express it as a formal
wff?

C-B

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

From: Charlie-Boo on 1 Jul 2010 10:51
On Jul 1, 9:52 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Charlie-Boo <shymath...(a)gmail.com> writes:
> > (Hmmm . . . If ZFC can't have a universal set, how can it have even
> > quantifiers??)
>
> A mystery!

I guess to both of us. Fancy that!

> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, dar ber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Charlie-Boo on 1 Jul 2010 11:11
On Jul 1, 10:04 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Frederick Williams <frederick.willia...(a)tesco.net> writes:
> > Look,
>
> >  (allX) |- P(X)
>
> > is hard to understand _whatever_ the domain of quantification (is that
> > what you mean by 'universal set'?).  Does it mean something like
>
> >  whatever term is substituted for the free variable X in P,
> >  the result is provable
>
> > or what?
>
> When the language under consideration has a name for every object in the
> domain of quantification -- as is the case with the language of
> arithmetic, which appears to be Charlie-Boo's concern here -- |- P(a)
> where a is an object in the domain is naturally taken to mean |- P(a*)
> where a* is the name for a. So
>
>  (all x) |- P(x)
>
> means that for all objects x in the domain of quantification the formula
> P(x*) is provable. This is standard convention in e.g. certain corners
> in proof theory where we don't want to get mired in tedious notational
> horrors simple-minded pedantry about the use-mention distinction might
> otherwise inspire. We encounter, for example, the uniform reflection
> schema written as:
>
>  ((all n) |- P(n)) --> (all n)P(n)
>
> There's no need to get one's knickers in a knot over this convention,
> provided all parties involved know what's going on. Alas, I very much
> doubt this is the case with our incorrigible

You mean to say that all my detractors are WRONG???

But be careful. People might catch on to your proclivity to throw
around impressive sounding words without actually doing anything with
them.

But as far as that goes, I noticed that you jumped in right when I was
asking the Conservatives how they define quantification and its
relationship with the universal set. That reminds me of Torkel, who
would admit he was wrong but blame it on his dealing with a system of
higher regard than what people were discussing, right when I was about
to prove him wrong. (Some of his last posts were of this type.)

> friend, that formidable
> formalizer of all of computer science.

Think of it as being a modality of Modal Logic.

And thanks for the ringing endorsement (of the system though strangely
not its author.)

Lastly, I noticed that the psychopaths whom I know are typically (1)
dishonest, (2) abusive, and (3) irrational. Your rare burst of
honesty is definitely a move in the right direction.

C-B

> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, dar ber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

First  |  Prev  |  Next  |  Last
Pages: 1 2 3 4 5
Prev: Cantor's First Proof
Next: math symbols in unicode collection