Conference rusure::math

    So maybe the problem is mistated, maybe the definition refers only to
    members of the null set.
    
    >Consider N+1. By definition, this number cannot be described with 15
    >words or less. However, I just described it with 15 words above: "The
    >largest integer number which may be decribed with up to fifteen words
    >plus one"
    
    I submit that "described" is an inadequate term to use in this context. 
    Try "specified" instead.  Or maybe any number can be specified with up
    to 15 words.  Certainly 2 can be so specified.  Then 2**2 can also! 
    and 2**(2**2) and 2**(2**(2**2)).  The specification of integers by
    words is not a well defined activity to which we can apply quantitative
    or exact logic.  Given that the set of which N is the largest member
    cannot be specified, can we specify its largest member?
    
    fwiw,
    
    Dick
    

re .0
    
>So, there must exist the largest integer number which may be described with up
>to 15 words. Call this number N.
    
    Think of this as a proposition regarding the existence of N to be disproved
    by reductio ad absurdum.
    
    Since your conclusion is a contradiction we may deduce that the
    proposition is false i.e. such an N does not exist.
    
    I can conceive of two reasons of why N would not exist. Either the set
    is infinite (or more accurately unbounded above) or the set is not well
    defined (so that its maximum is not well defined). I would opt for the
    latter.

   > I can conceive of two reasons of why N would not exist. Either the set
   > is infinite (or more accurately unbounded above) or the set is not well
   > defined (so that its maximum is not well defined). I would opt for the
   > latter.

    I would also vote for the seconds option (as .1 suggests, "describe" is
    not well defined). A similar known logic puzzle proves that the sky is
    green. Consider the following box:

        .--------------------------------------------------------.
        | (a) This box contains three statements.                |
        | (b) Exactly one of the statments in this box is TRUE.  |
        | (c) The sky is green.                                  |
        `--------------------------------------------------------'

    Let N = the amount of TRUE statements in the above box. N must be in
    the range 0 to 3. Let's eliminate some of the options: N cannot be 0,
    since (a) is TRUE. N cannot be 1, since in that case (b) is TRUE but
    we already know that (a) is also TRUE. N cannot be 3, since then all
    of the above are TRUE, including (b) - a contradiction.

    Therefore, we are left with N = 2. Obviously now (b) is not TRUE, so
    the TRUE statements are (a) & (c). Hence the sky is green (or purple,
    or orane, or anything else you want it to be).

  /Nitsan

    >    .--------------------------------------------------------.
>        | (a) This box contains three statements.                |
>        | (b) Exactly one of the statments in this box is TRUE.  |
>        | (c) The sky is green.                                  |
>        `--------------------------------------------------------'

    i like this. i think this comes from the ambiguity of (b) , if
    (b) was written like

    (b) Exactly one of the other statments in this box is TRUE.
                           ^^^^^

    i think the problem comes from (b)'s "Exactly one" has double 
    actions, it refer to itself and to others at the *same time*, which
    is not right, it say *one* but it could be taken to mean itself AND
    another one too.

    i think iam getting a headache from this, but it is a nice problem,
    i think logical problems like these are not good for the mind because
    they twist it up and then you have to un-twist it back.


    /nasser

    This is a more convoluted version of the classic:

	"This statement is false."

    to which is devoted a large literature.

				Topher

    >    .--------------------------------------------------------.
>        | (a) This box contains three statements.                |
>        | (b) Exactly one of the statments in this box is TRUE.  |
>        | (c) The sky is green.                                  |
>        -------------------------------------------------------'


I disagree with your logic, which I review before my argument:  You explained
how the number of true statements isn't 0 because... and it isn't 1 because...
and it isn't 3 because...

You then concluded that the number of true statements was THEREFORE 2.


That's where your logic fails.  What logical reasoning allows you to make
the statement that THEREFORE THE NUMBER OF TRUE STATEMENTS IS 2 ?

I would draw a different conclusion.  Instead of "therefore the number of
true statements is 2", I would go on to say "the number of true statements
isn't 2 because we know (c) is false, we know (a) is true, and hence (b)
is neither true nor false"

Hence I conclude that there is no defined number of true statements
in the box (since we've eliminated 0,1,2 and 3)

Your fallacy is actually right at the beginning, where you assume that
there is a countable number of true statements in the box at all !  I've
shown that there isn't any countable number of true statements.


/Eric

    re the second problem
    
>        .--------------------------------------------------------.
>        | (a) This box contains three statements.                |
>        | (b) Exactly one of the statments in this box is TRUE.  |
>        | (c) The sky is green.                                  |
>        -------------------------------------------------------'
    
    I would like to propose an alternative interpretation.
    
    "statement" is an undefined element here.  We have only our intuition
    to help us decide what counts as a statement.  a) and b) can only be
    verified or falsified if one has 
    
    	[1] a procedure for deciding what is a "statement" and 
    
    	[2] a procedure for deciding whether it is "true" or not. 
    
    but c) is empirically verifiable once we agree on what counts as "the
    sky" and what counts as "green".  So, the three statements are not
    subject to the same logic.  We have mixed our metaphors, so to speak,
    and should not be surprised that we get strange results.
    
    Dick

re. last:

This raises the issue of what a mathematical proof really is. Terms such
as "statement" are often used within "kosher" proofs. Is it right to use
the English language freely (or any other non-formal language) for proving
mathematical theorems? Especially when it comes to logic theory, we may
fail (without knowing) on symptoms resembling the last couple of "proofs"?

Who is to tell what is a right proof and what is not? More specifically:
It may be easy (?) to "describe" why certain proof is false (like previous
replies did), but how can one "validate" a proof?

Note that a common thing about the last few, is their way of self referencing
(sort of "Meta" problems). But that notion is not well defined, and may
also be present in valid proofs (?).

Late hours and good background music ("All About Eve") make you think.
/Nitsan

re: .-1

. Note that a common thing about the last few, is their way of self referencing
. (sort of "Meta" problems). But that notion is not well defined, and may
. also be present in valid proofs (?).

Didn't Goedel simply formalize this self-reference, make it well defined, and
in the process say something fundamental about all formal systems?

I don't think making these statements more "well defined" will get you out
of the hole.  You will still, in the end, have "This statement is false." to
deal with.

    re -.1, -.2
    
    Goedel did indeed formalize self reference and using that, prove some
    interesting things, among them, that not everything which is "true" can
    be proved.  But I think these problems are more basic and simple.  If
    we can't define our terms to mutual satisfaction, we don't know what
    we're talking about.  Defining terms (or deciding not to define them)
    is where we begin.  And one of the undefined terms in this string has
    been "statement".  That's all right, but then we also left undefined
    the relation between "statement" and "true", which means that (like a
    famous proverb) we neither know what we're talking about nor whether it
    is true.