Chapter 25 deals with the Set Theory; more specifically, it
deals with the concepts of aleph-null and aleph-one. Aleph-null and aleph-one
refer to subsets and types of infinity. Aleph-null, as implied by the term
“null” or zero, constitutes a smaller infinity than aleph-one. Yes there are
different sizes of infinity. The aleph-null sets are seen as finite sets of
infinite numbers. Examples of such sets are the following: rational numbers,
set of prime numbers, set of integers and even the number of atoms in the
cosmos. These sets of numbers are infinitely long but we see them as finite
infinite sets because, in theory, we can count all of these numbers. Aleph-sets
are infinite sets of infinite numbers, or uncountable infinities. Examples of
an uncountable infinity would be the square root of 2, pi. These numbers fall
in the aleph-one set so does that mean that the square root of 2 is a higher
infinity than all of the rational numbers? YES! The square root of two is an
irrational number, indicating that it has nonrepeating and never ending numbers
after the decimal, and seeing how we cannot reach the end of an irrational
number it is viewed as uncountable. The example shown in the book to
demonstrate the concept of aleph-one is similar to one that we did in class.
Lets say we take the “n” amount of numbers in between zero and one. Now lets
assume that these numbers are infinitely long and all different. Then let us
stack the numbers on eachother:
0.01223…
0.23451…
0.95674…
0.01234…
0.32718…
This list in theory goes on forever and can contain an
infinite amount of numbers. After you have the set of numbers written out, draw
a diagonal line from the first number, “N”, after the decimal of the first
number, then through the second number after the decimal of the second number
and the third decimal after the third number etc. Upon doing this you will get
a new number, according to the given set the number you get is 0.03638. Now
surely this number HAS to be within one of the numbers in our infinite set. No!
It can’t be within the numbers because the Nth number after the decimal will
always be different than a number in your set. Professor Trevino did a better
job of explaining this concept in class but I hope you catch my drift.
I had to
read this chapter a few times to finally wrap my head around the concepts of higher
infinities. I really enjoyed reading this chapter because it pushed me to think
at a more critical and higher level than I usually do. After learning about the
different concepts of infinity I was inspired to research this topic some more and
try and thinking of new categories and subsets of numbers that could qualify as
aleph-null or aleph-one. If you are going to read this chapter I would strongly
advise that you forget all that you think you know about infinity because you
will quickly learn, as I did, that you don’t know what you think you know.
I agree that this chapter was very hard to understand. Your summary definitely helped me understand it more!
ReplyDeleteI liked your summary of the aleph-null and aleph-one concepts. It really helped me understand the chapter more. I also agree that there is more to learn about infinity than we know.
ReplyDeleteYou had a really good summary and it helped me understand aleph-null and aleph-one better.
ReplyDeleteYour explanation of Aleph-null and Aleph-one sets gave me a better understanding of it. I do remember Professor Trevino teaching us about this concept of sets. It makes sense that the nth number after the decimal will always be different than a number in your set.
ReplyDeleteI found this chapter very interesting since my first blog post had to do a little with the aleph-null and null sets. And i definitely understand that the concept of "infinity" is hard to grasp, since I had a chapter that had a lot to do with infinity. Overall i thought you did a great job explaining "Aleph-null", and "Aleph-one".
ReplyDeleteI can't really add much to this summary. I thought it was well put together and very thorough. It helped me understand the chapter a lot better.
ReplyDelete