Math CentralQuandaries & Queries


Question from Justin, a student:

Hello, I know I have asked a similar question before but I was just wondering if set theory applies to the lim x->0, y=1/x=infinity and if so, what type of infinity would it be? Thanks a lot for your help with this question!




Although the word infinity is used in both situations they are entirely different concepts. As was pointed out in our responses to some of your earlier questions, Cantor used the concept of a 1-1 correspondence to compare the size of sets. In doing so he showed there is a hierarchy of sizes of sets. These sizes are called cardinal numbers and for infinite sets, infinite cardinal numbers.

To say the limit of 1/x approaches infinity as x approaches zero is just a succinct way to say that as x approaches zero the values 1/x are unbounded. That is for any real number M you can find an x so that M < 1/x.




Set theory "applies" to this concept in the sense that set theory can be used to construct the natural numbers, which can be used to construct the rational numbers, which can be used to construct the real numbers, which can be used to define the concept of a limit at infinity.

The concept of infinity used in calculus, however, has more or less nothing to do with the concept of a transfinite cardinal number. It has perhaps slightly more to do with transfinite ordinals, but this is not generally a productive way of thinking about it.



Hello again, I was just wondering from your previous answer,
why you said that the infinity in calculus has perhaps slightly
more to do with transfinite ordinals?

Because the first transfinite ordinal, "small omega" (ω), corresponds to the order type of the natural numbers, and these occur naturally as a subset of the reals, embedded in such a way that a sequence of natural numbers approaches small omega within the ordinals if and only if its image in the reals "approaches positive infinity".

Cardinals, which count unordered sets, do not fit in as naturally (if "naturally" means anything here, which is dubious!)

You have developed a curiosity about matters that need much more precise statements to be really answerable. You need to read and understand a good introductory set theory textbook - and use the notation that you will learn there to formulate your questions.

I am not sure if there is any good complete resource online. Google Books has partial previews of Peter Johnstone's and Paul Halmos's books on set theory. The Wikibooks set theory text is still far from complete.


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