What is the Extended Real Line and How Does it Help Explain Infinity?

[woodysooner]

I know i posted this to all the general physics area, but I want an answer by pure mathematicians also so please don't throw this out mentors, i love yall though, just trying to do your job...

OK, basically I would like a mathematican or just a smart mathematical person ot explain infinity to me, I mean i have had calculus I II and III and still infinity perplexes me. It seems abstract and that it can't apply to physical phenomena, but to me mathematicans are also so brillantly insightful.

So enlighten me please.

 

[robert Ihnot]

Well, there are various classes of infinity. The simplest kind is the counting infinity. We know that we can always count higher; that if n is an integer, then n+1 is also an integer. This leads to an abstraction: The set of all integers. Very few, or nobody has a quarrel with this kind of infinity. It is essential to induction: A) If it is true for proposition P(1), and if it is true that P(n) implies the truth of P(n+1), then it is true for all integers.

This also leads to things like lim(1+1/n) as n goes to infinity. This limit is e. You can easily experiment with that using a calculator. Let n=1, 10, 100, 1000...pretty soon you will see that it settles down toward a limit, that it has an actual value in the limit.

But then they are higher order infinities described by Cantor. This is a more argumentative matter. Cantor showed that the line has more points than they are integers and fractions. But that there is any phisical application of transfinite numbers, I have not heard of any.

Yet, as far as the counting infinity it shows up all the time in Physics through the Calculus.

 

[woodysooner]

what are transfinite numbers?

 

[Galileo]

Infinity is at the heart of real analysis. The limiting process deals with infinity.
Whenever you take a derivative or evaluate an integral, you're dealing with infinity.
Same goes for sequences and series (adding an infinite number of terms)
But you will never see the symbol $\infty$ anywhere, only epsilons and delta's. That is the way to deal with infinity.
Cardinal numbers form another way infinity shows up. The set of real numbers is infinitely more large than the set of rational numbers. The real numbers can themselves be defined by a limiting process.
Physicists use real numbers to denote most physical quantities. Meaning the range of values of (say energy, or position) form a continuum (at least in Classical Physics).
So in that sense, infinity is everywhere in physics also.

 

[zeronem]

This also leads to things like lim(1+1/n) as n goes to infinity. This limit is e.

Actually it is

$$\lim_{n\rightarrow \infty} (1 + \frac {1}{n})^n = e$$

You forgot to put it to the nth power

 

[TenaliRaman]

As for transfinite numbers,
http://en.wikipedia.org/wiki/Transfinite_number

 

[matt grime]

A semi-physical need for transfinite cardinal numbers:

Let A the category of quasi-choerent sheaves on a compact manifold, then D(A) its derived category exists in this universe: proof, take a cardinal bounding the cardinalities of the stalks at every point and the cardinality of the cardinality of the set of points of the manifold and do something very nasty and complicated.
Why is that physical? Well, you'd need to ask some theoretical physicists who have interests in central charges and TQFT or something.

Note, when people are saying "larger" they simply mean that whilst there is an injection from X to Y, there is no bijection, and hence that the cardinality (loosely, very loosely, the size) of X is less than Y. This confuses many people because they don't understand that this is simply a definition.

 

[robert Ihnot]

Actually it is

$$\lim_{n\rightarrow \infty} (1 + \frac {1}{n})^n = e$$

You forgot to put it to the nth power

In the latex code how do you find those weird Vs with the double line in them?

 

[Muzza]

You mean something like $$\mathbb{V}$$? In that case, click the graphic to see the code.

 

[Hurkyl]

If you head over to general physics, there's a LaTeX sticky thread that's good for practicing and getting help.

 

[mathwonk]

oh my. i just wrote a long post but it got lost somehow. perhaps netscape does not work right here? or can i not be logged in on both netscape and internet explorer?

 

[Ethereal]

How does one resolve Cantor's antinomy? Cantor showed that the cardinality of the power set of any set is greater than that of the set. But if S is the set of all sets, then its power set would be larger than it; a contradiction. Do we deny that set S can exist?

 

[matt grime]

Yes, the "set of all sets" is well known not to be a genuine set in models of the set theories we care about, though plenty of people try and invent axiomatized set theories that do have a universal set, though I know of no successful ones.

 

[StonedPanda]

For your calculus purposes (and perhaps for all purposes) infinity is just a shorthand way of picking arbitrarily large values for, say, x.

 

[Alkatran]

It's always bothered me that I'm essentially dividing by 0 to get the derivative of a polynomial in math. I accept it, I understand it, I just don't like the idea that:

0 = lim(0) != 0

 

[mathwonk]

read the definition of a limit, and of a derivative. it has nothing to do with dividing by zero.

 

[Alkatran]

read the definition of a limit, and of a derivative. it has nothing to do with dividing by zero.

Oh I know how and why it works, it's just always bothered me. You eliminate H while you're using lim(0) then change lim(0) to 0 in the next step.

 

[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:

$$\lim_{h \to 0} \frac{(x+h)^2 -x^2}{h} = \lim_{h \to 0}2x + h = 2x$$

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]

Yes, the "set of all sets" is well known not to be a genuine set in models of the set theories we care about, though plenty of people try and invent axiomatized set theories that do have a universal set, though I know of no successful ones.

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 $+ \infty$ and $- \infty$. We preserve the original order in R, and define

$$- \infty < x < + \infty$$
for every $x \in R$

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

For additon and subtraction
$$x + \infty$$ = $$+ \infty$$
$$x - \infty$$ = $$+ \infty$$
For divison
$$\frac x {+ \infty} = \frac x {- \infty}=0$$

for multiplication
if $x > 0$ then $x \cdot (+ \infty )= + \infty, x \cdot (-\infty) = -\infty$

if $x < 0$ then $x \cdot (+ \infty )= - \infty, x \cdot (-\infty) = +\infty$

 

[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.