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Name: Justin
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Question:
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 this reasoning?



Replies:
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.

Vince Calder


Justin,

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
Physics Instructor
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.

Conrado Lacerda



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