Chapter 26 Supertasks

Chapter twenty-six is an interesting chapter. It has many concepts and within these different concepts there are many rules which lead to make these concepts true. Throughout this chapter the concepts that were being taught were, “finite sets”, “subsets”, “null set”, “transfinite numbers”, “cardinal number”, “aleph-null”, and “Zeno’s Paradoxes”. These are many different concepts yet they are all relatable in a sense, and the way these concepts relate is this theory of “infinity”.

    Finite sets are generated by x elements. And the way subsets are generated are by the elements in the finite set. For example, “a set of three elements, ABC, has 2^3=8 subsets: ABC, AB, BC, AC, A, B, C, and the null set. This is in a way related to what we were learning in class with the different combinations we had on the board. The big difference though is the fact that there is a null set. A null set is defined to be nothing essentially being 0 in a sense. In that it will always be capable of being used in all sets. What I had also gathered from the subsets is that an element cannot be matched with itself because it cannot be included within its own subset. Definition of a transfinite number is that, “it is the number of subsets of n-must be a higher order of infinity than n.” The term “aleph-null” is typically used for the lowest transfinite number. “It is the cardinal number of the set of all integers, and for that reason is often called a countable infinity. Any set that can be matched one to one with the counting number, such as the set of integral fractions, is said to be a countable or aleph-null set.” The last term, “Zeno’s Paradoxes.” Both terms seek to show this continued pattern that is shown through the Zeno’s runner theorem. What takes place in this theorem is that a runner half the distance in half a minute, a fourth of the distance in a fourth of the time, then an eighth of the distance in an eighth of the time, and so on until he has reached a total of one minute in which then means he has reached the end point. Yet if this were to happen this means that the runner were to have to run at the same pace throughout the entire minute until he has reached the final point.

    This chapter was very interesting but also very confusing in which made it frustrating to read at moments. There were many concepts which made me have to re-read at moments because of there being multiple concepts it caused there to be some confusion when matching the definitions to their concepts. Plus when first reading this chapter it was first hard to understand what i was being taught because I had never heard about a “Null set”, but I kept re-reading and asked for an explanation which then led me to find out that this chapter is interesting and in some ways relates to what we are being taught in class.