> > > > > > ALL(F):N->R  E(r):r  ALL(n):N  F(n)=/=r

> > > > > I'm not trying to trim to aviod anything other than the brokeness

> > > > > of the google groups.

> > > > > I'm trying to understand what you wrote.  It seems like you

> > > > > keep changing your symbolic language.

> > > > > Please verify:  For all functions of a natural returning a real

> > > > > there exist a real such that for all natural numbers the function

> > > > > will not return that real.

> > > > > I believe that was his conclusion.

> > > > > I don't know how to write in that language yet.  Does this make

> > > > > sense:

> > > > > EXISTS(G):F(N),N->R ALL(F):N->R ALL(m):N ALL(n):N F(n)=/=SUM(G(F(m),m))

> > > > G is just NXR->R

> > > > (string-of-digits wise function)

> > > > or

> > > > {0..9}->{0..9)

> > > > (digit wise function)

> > > I'm not sure I understand.

> > > G takes then nth value of F and returns a real between

> > > .1^n and .1^(n+1) that differs from MOD(FLOOR(F(n),.1^(n+1)),.1^n)

> > > in a way where it will never be 9*(.1^(n+1)).

> > OK but you and George and all logicians are making the same mistake.

> I'm a computer programmer/systems analyst.  I was deeply depressed when
> I took my symbolic logic classes so don't remember them very well.  I'm

> trying to relearn them as fast as I can.  It's hard to do so given the
> diverse symbolic systems being used here and very bad assertions about
> the truth of the matter.

> Constructivist,http://en.wikipedia.org/wiki/Constructivism_(mathematics)
> make it very hard to talk about infinite sets.  I never let my mathmatics
> be limited by the computers I've used and I never will.  I do not accept
> there is no number if the system using numbers cannot encode it in native
> mode.  Integers weren't limited to 256 just because that's all my 8080
> could handle.

> > ALL( FUNCTION ): DOMAIN->RANGE
> > ALL( FUNCTION OF FUNCTIONS )  DXR1 -> R2
> > for FUNCTIONS, i.e. LOGIC FORMULA  use  "D->R"
> > for SETS, i.e. ordered pairs use "DXR"

> Can you point me to a wikipedia page or .edu page
> using the syntax you are describing.  I'd like to

> come up to speed as quickly as possible.  I'm not
> sure I agree to these relative to what I intend to
> write.  I take D to reference "domain".  Do you take
> it as a given that an ordered pair includes a natural
> number as one of the items paired. where DXR would
> reference an ordered pair (N,R)?  Would you use DXN
> if the pairing was (N,N)?

> > then you can safely make claims about super-infinite sets in FIRST
> > ORDER LOGIC about

> >   SETS:DXR

> > without stuffing up the
> >   FUNCTION:D->R
> > specs for us programmers.

> I keep reading what you write about microProlog and wff.
> I still disagree with your formulation since the compiler
> necessarily doesn't allow anything but wff into the system in
> the first place.

> What does:

> ?wff(this(

> return?  Does it return no?

> where <= is being used as "less than or equal to".
> I'm leaving the brackets in place because it appears
> some use it as a transform from true to 1 and false to 0.
> I dont get it in this context.  It seems to mix some
> programming languages' coding for the comparison operators
> with their logical value.

> A(n)  0 <= n
> A(m) A(n) s(m)<=s(n) -> m<=n

> > F is just a LIST!

> Cantor deals with a well ordered list.

> The bijection has to be done with a well ordered list.
> Cantor produced a well ordered list of reals.  What makes
> a list well ordered isn't that the elements are in numeric
> order but that one of the elements is the first item on
> the list and all of the items on the list other than the
> first has a uniquely identified successor based upon their
> position on the list.  In reality the position in the list
> alone identifies uniqueness not the element at that position.
> A problem can arise if the elements are not uniquely identified
> because the positional relationship <= (less than or equal to)
> can't be verified, for instance, giving the well ordered list
> {a, b, a, c} one cannot say a<=b without identifying which a
> one is talking about.

> > LIST ITEM 1  =  REAL#1  =  0.322323...

> f(0) = 0.322323... = 0.32[23]
> f(1) = 1/3
> etc. is just fine for a function as long as all of the items
> on the list are uniquely associated with a set of natural numbers
> in their defined successor order starting with 0 and all natural
> numbers in the set are uniquely associated with the elements in the
> list.  The

>    ALL(F)N->R

> part says for all orderings of lists of reals...

> > > > > > > I believe that was his conclusion.
> > > > > > > I don't know how to write in that language yet.  Does this make
> > > > > > > sense:
> > > > > > > EXISTS(G):F(N),N->R ALL(F):N->R ALL(m):N ALL(n):N F(n)=/=SUM(G(F(m),m))
> > > > > > G is just NXR->R

> > I'm just saying your TYPE of G is not well formulated.

> Yes, I get that.  I'm trying to restrict the domain of G to
> the pair of a particular function whose domain is a
> set of natural numbers and whose codomain is a set of reals
> and the particular natural number used to produce the particular
> real in the set of reals produced by the function.  Details are
> important.

> > If you are summing the digit positions into d1/10 + d2/100 + d3/1000

> > + ....

> > that's fine!

> Yes, that's what I'm doing.

> > If you want to put ALL FUNCTION at the start you are talking about a
> > bona-fide function here not my countable LIST:N->R of domain N.

> By saying the list is countable you are implying a function can
> be created to count them.  I want to talk about all such possible
> functions for all orderings of the list.  Certainly a countable
> list will have the same members no matter what order it's in.
> A function that finds a real not in the list in one ordering
> must not exist in a different ordering otherwise they wouldn't
> contain the same elements.

> ...

> > That way I can do LOGIC in PROLOG.  Actual LOGIC!  And Backtrack from
> > Normal Clauses to Horn Clauses, theorem proving capability PLUS axiom
> > back tracking derivation capability in the 1 system.  Normal Clauses
> > OVER Horn Clauses.  This is just MY Method but it looks like it would
> > have general Utility.

> My issue is your use of WFF.  It's non-standard.  A formula is
> well formed by its form not its logical state or lack there of.
> By not using the standard definition your conclusions are off
> and this reduces the utility of your work.

> I've been trying to relearn prolog as well so I can address some
> of the issues.  I've had to move here because google doesn't allow
> cross posts any more and you don't read comp.ai.philosophy even though
> you add a cross post to it.

> As it turns out I learned prolog on a (now defunct) Bordland product
> that appears to be somewhat related to Visual Prolog.  That version
> of Prolog had string functions.  I could parse BNFhttp://en.wikipedia.org/wiki/Backus%E2%80%93Naur_Form
> to define a language's syntax then bind elements as needed to lists or
> functions.  It was a lot of work and the limitations of my patience
> lead me to move on.  Unix programmers have created lots of useful
> scripting languages to aid LALR compiler generation.

> http://en.wikipedia.org/wiki/LALR_parser