Infinity to a Mathematican

matt grime

That should only bother you if you don't understand the the arithmetic operations are continuous. And you're only putting zero in because you can prove it behaves properly, for instance, to show x^2 differentiates to give 2x, are you bothered by:

[tex]\lim_{h \to 0} \frac{(x+h)^2 -x^2}{h} = \lim_{h \to 0}2x + h = 2x[/tex]

if so then there is no reason to since it is a basic and easy proof that lim(a(h)+b(h)) = lim a(h) + lim b(h), for a, b functions of h, as h tends to h_0, when ever the individual limits exist.

TenaliRaman

Alkatran,
[question]
i think this is ur question,
lets take the same example of x^2
during the simplification of [(x+h)^2-x^2]/h
we cancel out h's from the numerator and denominator saying that
"since h->0, h!=0"
then in the subsequent step we proceed to put h=0 (while in our previous statement we say that h!=0 ... a contradiction ???) and find the limit.
[/question]

Ofcourse the key is to look at the definition of the limit more closely and see why do we say h=0 (this is usually an engineer's method .... someone who tries to find the easiest way to find the answer .... ofcourse this has justification!!) and that it gives us a limit(!! can i call it a bound ?? can i find one which is better ?? some questions to ponder).

-- AI
P.S -> someone who is not aware of the justification can find it hard to accept this "general engineer's technique" as i call it, i would advise to start finding the limits the way it is defined ... soon one sees the light ..... if u can, do refer "Thomas and Finney for this one .... they really do shed some light on this aspect (IIRC).

ZeroZero

Very interesting thread... for all its worth, try the book "Everything and More: A Compact History of Infinity" by David Foster Wallace... it is a "popular science" kind of book on the topic... great read, doens't offend your intelligence and the bibliography is superb.

fourier jr

i didn't there could be a set of all sets because that would lead to russell's paradox. maybe i'm thinking of something else

Integral

This is a thread about what infinity is to a mathematician. why has no one yet posted the mathematical definition of infinity in the Real number system.

From Rudin, Principles of Mathematical Analysis

Definition: The extended real number system consistes of the real field R and two symbols [itex]+ \infty [/itex] and [itex]- \infty [/itex]. We preserve the original order in R, and define

[tex] - \infty < x < + \infty [/tex]
for every [itex] x \in R [/itex]

The extended Real Numbers do not form a field so we must define the results of various operations.

For additon and subtraction
[tex]x + \infty [/tex] = [tex]+ \infty [/tex]
[tex]x - \infty [/tex] = [tex]+ \infty [/tex]
For divison
[tex] \frac x {+ \infty} = \frac x {- \infty}=0 [/tex]

for multiplication
if [itex] x > 0 [/itex] then [itex] x \cdot (+ \infty )= + \infty, x \cdot (-\infty) = -\infty [/itex]

if [itex] x < 0 [/itex] then [itex] x \cdot (+ \infty )= - \infty, x \cdot (-\infty) = +\infty [/itex]

fourier jr

those might be "definitions" given in intro analysis books, but i think those are just conventions. i would have thought that the definition would include something about infinite sets and the fact that there is a bijection between an infinite set and proper subsets of it. (like the integers & even integers, or something like that).

Hurkyl

Nah, they're actual definitions. The extended real line is a compactification of the ordinary real line. The definitions of the arithmetic operators are then extended continuously to the two new points.

It's a very useful topological space, because it replaces the notion of "diverging to infinity" that you use in the ordinary real line with honest-to-goodness convergence in the extended real line.

And, of course, this topological construction of the two "points at infinity" has nothing to do with the notion of cardinality and infinite sets.