Re: abundance of irrationals!)

*** T. Winter said:
> In article <MPG.1ce41773427854f7989bc0@xxxxxxxxxxxxxxxxxxxxxxxxx> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> ...
> > I don't tend to deal with these axioms much. The only reason I am dealing
> > with them now is that mathematicians seem not to be able to do without
> > them.
>
> Well, you know, without axioms, no mathematics... You have to start with
> some basic truths, otherwise you can not logically reason to conclusions.
> The basic truths are the axioms.
Axioms are not "basic truths". The axioms are assertions which, within the
logic of the axiomatic system are atomic statements assumed to be true, and
treated in that system as "basic truths". HOWEVER, the "basic truths" are NOT
the axioms. The axioms are statements that we agree are true, based on reasons
OUTSIDE of the particular axiomatic system.

This is true when dealing with logic in general. We start with a certain set of
given "facts", each with a certain truth value (generally 1, or 100% true), and
a logical statement using those atomic facts, and evaluate the truth value of
the logical statement, by plugging in truth values and evaluating the logical
construction. Now, if all our facts are true, and all our logical operations
are correctly performed, we will get an correct answer. IF, however, any of our
assertions is FALSE, then no matter how well we perform our oeprations on the
truth values of those "facts", we are going to get an erroneous result.

Axiomatic mathematics IS logical inference. ASSUMING all the axioms we have
asserted are CORRECT, then we can prove a given logical statement involving
them. If we get results that make no sense, then we can examine our logic to
see if we made a mistake. If there is no mistake in the logic, then the
erroneous result MUST be the result of one or more false "facts", axioms, or
other logical assumptions, which sometimes go entirely unstated. When this is
the case, and we must examine the axioms for validity, we must examine the
justifcations for these axioms, which by definition lie OUTSIDE of the
logical/axiomatic system we are examing. If they were derived from other axioms
within the logical framework, then they would be theorems.

The truth of axioms can be tested, as I said before, by their compatibility
with all other axioms and their compatibility with reality. They can be
examined from the other end by examining exactly why we asserted that axiom to
begin with. Usually the justification of an axiom rests on some kind of logic
involving assertions outside the system we are examining, and in some sense
represent theorems of that external system, which rests on its own axioms. In
order to not have to include those external axioms as axioms in our system, so
we can keep things simple, we express the external theorems atomically as
axioms within our system. Otherwise we get confused. But this can also lead to
confusion, when we forget the derivations of our axioms when we run into
contradictions, because it is always a possibility that the justification for
an axiom is flawed in its derivation external to the axiomatic system at hand.

>
> > > >> For example, the (correct) statement
> > > >> the limit of f(x) = x sin( 1/x ) as x tends to 0 is 0.
> > > >> may be expressed as a cumbersome but precise statement about the
> > > >> behaviour of the function f near zero. The words "end" or "stop"
> > > >> never occur in the translated statement.
> > >
> > > >near zero, or at zero? Most limits give pretty precise values for what is
> > > >happening at that precise point.
> > >
> > > Oh? What is "happening at [the] precise point" x=0 for this function f ?
> > > Are you saying the limit at x=0 does not exist because f(0) is undefined?
> > Well f(0) is undefined, why? Because sin(1/0), or sin(oo), is undefined. But,
> > we know that sin is always between 1 and -1 and so x sin(1/x) at x=0 is
> > equivalent to 0*(some number between 1 and -1), which is always 0.
>
> Except when you go to the complex numbers, where sin(x) can be any value,
> including values larger than 0... But what you are telling above seems a
> lot like taking a limit...

It's not dissimilar, but doesn't require the full extent of limits. It reminds
me of a story....(dreamy music as the newsgroup swirls and fades into a picture
of my old place, a converted tool shed, as far as I can tell....)

I was playing cards with a buddy, five card draw. I drew a 6,9,J,Q,K. Ooohh,
all I need is a 10, so I'll throw the 6, and get an almost royal straight! I
didn't get a 10, but the story is good anyway. When I said it was worth a try,
since I had a 4/47 chance my friend didn't believe me. It couldn't be that
high. (it's barely over 1/12???) So, I said, "I have five cards I know of, none
of them are 10's, so of the 47 cards I don't know about, from which I about to
draw 1, 4 will satisfy my needs. Hence 4 chances in 47." Now, he just couldn't
believe it was that simple. He insisted that he might have all four 10's, in
which case I had NO chance. Pointing out that that was highly unlikely didn't
sway him. If he had even one 10, then I'd only have three chances, etc. So, in
order to prove it to him, I had to break the problem into five possibilities,
that he had 0,1,2,3 or 4 10's, calculate the chances of each of those
possibilities happening, and then calculate, given each of those possibilities,
what my chances were of drawing a 10 from the 42 cards (or had he already drawn
- doesn't matter) that were actually in the deck to choose from. Then, for each
of the five possibilities, I had to multiply his chance of having that number
of 10's, by my chances of getting a 10 given his possession of that number of
10's, and then add up the five products. Guess what the sum was? 4/47. Two
pages of scratches for what was figured out in two seconds by other means. This
is not a dissimilar experience.
>
> > > To me there is no confusion: f is not defined at x=0. On the other
> > > hand the limit of f, as x tends to zero, is indeed L = 0 because
> > > for every _nonzero_ x, |f(x)| <= |x|, which means that the values of f
> > > can be kept smaller than any pre-assigned epsilon simply by keeping
> > > x close to (but of course different from) epsilon.
>
Somehow, the above looks indented like my statement, but I don't think it is.
> But this variant also works for complex x, if you change "_nonzero_ x" to
> "_nonzero_ x with |x| < 1".
>

--
Smiles,

Tony
.