Re: abundance of irrationals!)
- From: rusin@xxxxxxxxxxxxxxxxxxxxx (Dave Rusin)
- Date: 4 May 2005 21:42:50 GMT
Tony has more gripes than I can deal with today but he asked for the
mathematical definition of two terms and since I have students' topics
on the brain this week I will oblige him on both.
In article <MPG.1ce2ec19ab7de659989bb6@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>Dave Rusin said:
>> mathematicians might prefer to treat "infinite X" as a
>> single noun phrase to be defined
>Is there an agreed on definition of "infinite" that enables us to know what
>each other is talking about?
We say a set A is infinite if there is an injection A --> A which is not onto.
(That defines "infinite set". It doesn't define "infinite elephant"; I don't
know what that would mean. It doesn't define "infinite integer"; I don't know
what that means either, though I can usually try to get the gist of what
people mean when they use it -- they mean something which is not, in fact,
an integer, but that sort of thing may be permitted among friends! After all,
a "red herring" is not red nor a herring -- it's a figure of speech.)
Some people do it differently: they say the previous definition should
be of the phrase "Dedekind infinite", and then they define a companion
phrase like, say, "Cardinal infinite" as follows:
Definition 1. A set A is Cardinal-finite if there is a bijection
A --> { 1, 2, 3, .., n } for some natural number n .
Definition 2. A set A is Cardinal-infinite if it is not Cardinal-finite.
Then you get to do some interesting mathematics:
Theorem 1. If a set A is Dedekind-infinite then it is Cardinal-infinite.
Theorem 2. If a set A is Cardinal-infinite then it is Dedekind-infinite
if the Axiom of Choice is accepted.
(That is, the proof of Theorem 2 uses the Axiom of Choice. Is the Axiom of
Choice "true"? The answer to that will be the same as the answer to the
question, 'Is Theorem 2 "true"?')
Most people are happy to accept the axiom of choice, which means we have
a pair of theorems here which states that a set is Cardinal-infinite iff it
is Dedekind-infinite, and sets which have either (hence both) property are
then just called "infinite". Probably if you're in the camp that does not
use the Axiom of Choice then you use "Cardinal-infinite" as a synonym
of "infinite" and merely discuss "Dedekind-infinite" sets as curiosities.
I wouldn't really know, not being in that camp.
>> 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?
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.
See? I told you it was cumbersome -- and that's the informal version.
The most formal version is:
A e >0 E d >0 ( ( |x| < d and x not 0 ) => ( |f(x)| < e ) )
which would be read as
For every positive epsilon there is a positive delta such that for all
nonzero x whose distance from 0 is less than d we have that f(x)
differs from 0 by less than epsilon.
No "end". No "stop". No "happening at x=0". And while I'm at it let
me note that I chose this f as my example because there is no
"infinite" either, and because there's no "...but never equal to"
either, in the sense that I most definitely DO have f(x) equalling
the limiting value L = 0 over and over again (namely for each of
x = 1/pi, 1/(2pi), 1/(3pi), ... we have f(x) = 0 ). More details,
including definitions, motivations, examples, and theorems, can be
found in any decent calculus book.
This may or may not correspond to your intuitive notion of "limit".
If it does, fine. If not, then please don't use the word "limit"
when referring to the behaviours of functions as the inputs approach
a specific point a (or as the inputs increase without bound) because
that would cause confusion. Define YOUR notion precisely and then
call it something else -- you can refer to the "Orlow-limit of f at a"
or the "tendency of f near a" or as "L*( f, a )" or whatever.
Make sure your definition is clear enough so that one can deduce
from the definition what (e.g.) L*( x sin(1/x), 0 ) is supposed to equal.
You don't need to agree that mathematicians' language is appealing.
You should know, though, that it is incredibly universal. On the
rare occasions when someone will offer definitions notably different
from mine, s/he will be able to prove that the definitions are
logically equivalent to mine. This is sci.math here so I think it
is quite reasonable to insist that common math terms not be used to
refer to non-standard concepts; pick new names for those new concepts.
(E.g. "natural numbers"="set satisfying Peano axioms", "real numbers"=
ordered field with LUB property, etc.) If you're not sure what the
math terms are, read a book (<-- best choice!) or ask here and some
helpful person will answer a polite query.
dave
.
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