Re: abundance of irrationals!)

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.

> > >> 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...

> > 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.

But this variant also works for complex x, if you change "_nonzero_ x" to
"_nonzero_ x with |x| < 1".
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