Conference rusure::math

>> Can it be shown that there are no ineffable numbers by an argument that begins
>>
>> "Let I be the smallest positive ineffable number...."
        
        No.  The ordering of the reals is such that a nonempty
        set of them need not have a least element.  Otherwise,
        why not show that there are no rational numbers by
        arguing
        
        	"Let q be the smallest positive rational number.
        	 Consider q/2.  0 < q/2 < q so q/2 is irrational.
        	 But then q = 2 * (q/2) is also irrational, a
        	 contradiction."
        
        The incorrect assumption is not that rationals exist,
        it's that a least positive rational exists.  Likewise, 
        in a proof starting with the assumption "Let I be the
        smallest positive ineffable number...." the incorrect
        assumption (if any) would be that the positive ineffables
        have a least element; not that there aren't any.
        
        Dan

    The "joke" posting in 924.28 was a humerous reply to a "serious"
    posting, so you are welcome to take it seriously, if you wish.

    Observation 1: As defined there are no ineffable numbers, since you can
    find *some* language which can name any proposed ineffable number.
    Just let the language be, for example, standard algebraic notation
    with the addition of the special symbol �.  That symbol represents the
    given ineffable number.  What I think the poster was trying to get at
    is something like "any language interpretable by a Turing Machine" or
    some such.  In that case what he was probably invented and named was
    the set standardly refered to as the non-computable reals.

    Observation 2: The answer to your qeustion is that you assume that
    there is such aa thing as the smallest positive ineffable number.  But
    there is no such thing any more than there is a smallest positive real
    or a smallest positive rational (note: almost all reals are
    "ineffable", since there are countable "effable" (i.e., computable
    real) numbers but an uncountable number of reals).

    For any positive ineffable the value obtained by dividing it by 2 is
    also ineffable.  If it were not then there would be a "name" for the
    latter in some language, say �, which means that there would be a name
    for the former in that same langauge, to wit 2�.  But the latter number
    is smaller than the former, so that for any positive ineffable we can
    always find a smaller positive ineffable.  The set of positive
    ineffables is open at zero.

    Challenge: prove (or disprove) that between any two effables
    (computable reals) there is at least one ineffable. (Proving that there
    is an ineffable between any two ineffables, and one between any effable
    and any ineffable is trivial).

			    Topher

.1>        No.  The ordering of the reals is such that a nonempty
.1>        set of them need not have a least element.  Otherwise,
    
    But this applies only to the standard "<" ordering operator.
    The Well-ordering Theorem states that for ANY set S there is an
    ordering relation satisfying the property that EVERY nonempty subset 
    of S has a least element.  I suspect Wally had such an ordering 
    in mind when he wrote .0
    
    The Well-ordering Theorem is equivalent to the Axiom of Choice,
    so if you want to be a prude about it you can deny this line of reasoning.
    But there are better ways to refute the argument, and I think Wally
    was looking for some other arguments.  Otherwise, all numbers 
    are interesting, even 1729.
    
      John

>    But this applies only to the standard "<" ordering operator.
>    The Well-ordering Theorem states that for ANY set S there is an
>    ordering relation satisfying the property that EVERY nonempty subset 
>    of S has a least element.  I suspect Wally had such an ordering 
>    in mind when he wrote .0

    To apply this, however, you must specify/describe a specific ordering.
    The ordering would have to be computable.  The Well-ordering Theorem
    does not specify that the ordering is computable.  You would have
    to say "The smallest ineffable number relative to the relation �" and
    to say that you would have to have a "name" for � in the language.

    If we restrict "all languages" to languages with a computable semantics
    in terms of the real numbers (i.e., ineffables = non-computable reals),
    then we would have to prove that a computable relation exists which
    induces a well ordering on the ineffables to apply this argument.
    Anyone have a simple proof of this?  (Note: it is *not* necessary that
    you be able to compute the smallest element; only that given two
    ineffable numbers that you be able to compute which one is "smaller"
    and be able to prove that this computable relationship induces a
    well ordering.  And of course, you need only prove that such a
    computable relationship exists -- you do not have to present it).

    If we do not restrict what we mean by "all languages" the demonstration
    that the set of ineffable numbers is empty is trivial, as I said
    earlier.  We don't need to invoke the Well-ordering Theorem (a formal
    version of my previous argument, however, might require the Axiom of
    Choice, but I don't think so).

    There may be other ways to restrict "all languages".  The one that
    occurs to me is that names must refer to computable predicates with
    unique values for which they are true.  The question that the original
    poster raised on sci.math, however, implied that this was not what he
    had in mind.  That there is a computable predicate which is true for a
    unique real number does not obviously imply that there is a computable
    procedure for finding that unique real number.

					Topher

	Oops, another reply got in that overlaps my reply to .3,
        but what the heck, I'll post it anyway.
        
	re .2,

>>    Challenge: prove (or disprove) that between any two effables
>>    (computable reals) there is at least one ineffable. (Proving that there
>>    is an ineffable between any two ineffables, and one between any effable
>>    and any ineffable is trivial).

	Assuming that ineffables exist as described, let X be an
	ineffable

		zero is obviously effable, so X is positive or negative

		|X| is positive and still ineffable

		there is a strictly decreasing sequence of positive
		ineffables |X| > |X|/2 > |X|/3 > ... > |X|/n > ...
		with greatest lower bound zero.

		an effable plus an ineffable must be ineffable

		if E and F are effables, E < F, then for all
		sufficiently large n the number E + |X|/n is an
		ineffable between E and F

	re .3,

>> .1>        No.  The ordering of the reals is such that a nonempty
>> .1>        set of them need not have a least element.  Otherwise,
>>
>>    But this applies only to the standard "<" ordering operator.
>>    The Well-ordering Theorem states that for ANY set S there is an
>>    ordering relation satisfying the property that EVERY nonempty subset 
>>    of S has a least element.  I suspect Wally had such an ordering 
>>    in mind when he wrote .0

	Go ahead, invoke the Axiom of Choice.  So you have a set of
	well-orderings of the ineffables, and that set is nonempty.  Now
	how do you name a specific element of that set?  If you can't,
	you are just doing what Topher mentioned in .2, adding a special
	symbol to respresent "it".

	If you assume V = L or, less strongly, that all reals are
	ordinal definable, then you can define ("name") such a
	well-ordering.
        
        Dan
        

Someone back there in reply-land said:

	There is no smallest positive real.

Perhaps this can be improved in light of current discussion.  Let me conject:

	There IS a smallest positive real.  But it's ineffable, so don't
	ask me to present it.

I thought of this sort of in jest, but maybe there's something to it ?

/Eric

RE: .6 (Eric)

    Nope.

    If by the "smallest" you mean the smallest using the conventional
    ordering of the reals, than for every ineffable number there is a still
    smaller *effable* number (though I can't say which of the countable
    numbers I can "name" it is until you "give" me the effable, or at least
    some information about it).

    If you are talking about special "well-founded" orderings of the reals,
    than some are bounded below by effables and some by ineffables.  In
    fact, give me any well-founded ordering on the reals and any specific
    real, effable or ineffable, and I'll give you a well-founded ordering
    with that specific real number as the smallest element.  (In fact,
    given any complete ordering of the reals, and any specific real, I'll
    give you a complete ordering with that real as the smallest value).

				    Topher

    re.  <<< Note 1419.3 by VMSDEV::HALLYB "The Smart Money was on Goliath"
    
    Presumably the reference to even 1729 being interesting was to do with
    the tale Hardy tells in his book "A Mathematician's Apology" about his
    visit to "Every natural number was a personal friend" Ramanujan, on his
    deathbed.
    
    Hardy: (Looking uncomfortable and struggling for something to say)
    Well, I can't think of anything interesting about the number of the
    taxi I took to get here.
    
    Ramanujan: Really? What was it?
    
    Hardy: 1729
    
    Ramanujan: Nonsense! That is the smallest number which can be expressed
    as the sum of two cubes in two different ways!
    
    That was lightest moment in an otherwise very downbeat book. I much
    preferred the equivalent written by his partner Littlewood - A
    Mathematician's Miscellany with its calculations of "A snowball's
    chance in hell" and other such topics.
    
    Brian

>    Presumably the reference to even 1729 being interesting was to do with
>    the tale Hardy tells in his book "A Mathematician's Apology" about his
>    visit to "Every natural number was a personal friend" Ramanujan, on his
>    deathbed.

	I didn't know that it was Ramanujan's deathbed.   I thought he was
just ill.   If I had been R. in my final scene, I would have been unable to
resist going out with the gag:

	"Kiss me, Hardy."
    
Regards,
Andrew.

	Perhaps I should explain for non-Britishers, that "Kiss me, Hardy"
are the famous last words of Admiral Horatio Nelson, as he lay dying on
HMS Victory, knowing that he had won the Battle of Trafalgar.   They were
spoken to a certain Captain Hardy.

Regards,
Andrew.

>    That was lightest moment in an otherwise very downbeat book.

	This reminds me of a haunting story, which I believe to be true, 
laced with Freudian overtones and some delicate wordplay.

	A famous mathematician had a son out of wedlock, but decided in the
end to stay with his wife.   He gave the child a copy of "A Mathematician's
Apology", inscribed with the message:

	"For my son, when he is old enough to understand."

Regards,
Andrew.

>    That was lightest moment in an otherwise very downbeat book.

	This reminds me of a haunting story, which I believe to be true, 
laced with Freudian overtones and some delicate wordplay.

	A famous mathematician had a son out of wedlock, but decided in the
end to stay with his wife.   He gave the child a copy of "A Mathematician's
Apology", inscribed with the message:

	"For my son, when he is old enough to understand."


>    I much preferred the equivalent written by his partner Littlewood.
    
	I thought his partner was called Laurelwood :-)

regards,
Andrew.

re .10

>	Perhaps I should explain for non-Britishers, that "Kiss me, Hardy"
    
    I thought he said "Kismet, Hardy" where, if my shaky memory serves me
    correctly, "kismet" means "fate" in some language (not English). I stand
    to be corrected though.

>    I thought he said "Kismet, Hardy" where, if my shaky memory serves me
>    correctly, "kismet" means "fate" in some language (not English). I stand
>    to be corrected though.

	Well, no one really knows.   I don't think Nelson was giving elocution
lessons at that moment in time.   But certainly sailors are a comradely lot...
:-)   I think the Kismet story seems more plausible, but I decided that to
address it would be to enlarge the rathole unnecessarily.   The author of
-.1 however seemed keen to burrow further anyway.

Andrew.