Size of Infinity
I was just wondering if the infinity obtained by limits in
calculus is the largest possible quantity in mathematics and
calculus for the following reason. I think that the infinity
obtained in calculus (ex.lim x->0, y=1/x=infinity) or any other
limit that goes to infinity in calculus is the largest quantity in
mathematics and calculus because you cannot get a larger quantity
than this infinity unlike set theory where you can get a larger
infinity than the one before in sets. Thus, since the rules of set
theory do not apply to it and so you cannot get a larger infinity
than the one in calculus, I was just wondering if this is indeed
true that the infinity in calculus is the largest possible quantity
in mathematics since you cannot get a larger quantity than it with
Infinity is a difficult concept to get across. It is non-intuitive. There is no
number greater than infinity. That's easy if you tell me a number "N" I can
always respond with "N+1". There are infinities that are larger than other
infinities, i.e. more densely packed. There is no "largest possible quantity".
There are a number of books that consider the non-intuitive concept of infinity.
That's where you should begin to grapple with this concept.
Infinity is not the largest size you can get, because you cannot ever
really get it. Infinity is larger than any value you can really get.
This is why infinity can be said to equal infinity plus one. Both are
larger than you can ever get, so both are infinity.
Dr. Ken Mellendorf
Illinois Central College
Yes, your reasoning is correct, but some considerations are in order. First of all, the
term "largest possible quantity in mathematics" is too strong to be correct, and we need
to reformulate it. Since Calculus is essentially about real numbers and functions on the
real line, we could say something like "infinity is the largest real number".
Unfortunately, this is also incorrect, because infinity is not a real number (have you
heard of the Archimedean property of the real numbers?). What is usually done is to
add two more elements to the set of real numbers (the infinity and the minus infinity)
and to define how to operate them with other real numbers and between themselves. For
instance, if we multiply the infinity with a positive real number we obtain the
infinity, but if the real number is taken to be negative the result is minus infinity.
The set of real numbers together with infinity and minus infinity is called the
"extended real numbers system", and in this set infinity is indeed the largest
element. The extended real numbers system is very useful in many fields of Mathematics,
such as Measure Theory (since a measure is a function that relates to certain subsets
of a set a positive extended real number) and Topology (the extended real numbers
system is a compactification of the real line), but has a rather poor algebraic
structure, since infinity has neither and additive nor a multiplicative inverse, and
we cannot add the infinity to the minus infinity. It is also worth to mention that
all the functions that converged to infinity at some point have well-defined limits
in the extended real numbers system.
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Update: June 2012