Chapter 25: Aleph-Null and Aleph-One

Chapter 25 focuses on set theory. Set theory is described as “the branch of mathematics that deals with the formal properties of sets as units and the expression of other branches of mathematics in terms of sets” (Google). It begins with Paul J. Cohen of Stanford University finding an answer to what was defined as “one of the greatest problems of modern set theory” (Gardner 327): Is there an order of infinity higher than the number of integers but lower than the number of points on a line? Georg Ferdinand Ludwig Philipp Cantor defined this infinity of the integers are aleph-null or aleph sub 0. An example that Gardner used for aleph-null was given a set counted 1,2,3 and so on and put in prime numbers in one-to-one correspondence with the integers. The prime set is an aleph-null because it is countable. Another example that was used was the method that Charles Sanders Peirce created. Starting with the fractions 0/1 and 1/0. If you add the numerators and denominators of the two you would get 1/1 and that would go in between the starting fractions. If you repeat this, you would continually put the new fractions in between either the 1/0 and 1/1 or the 1/1 and 0/1. If continued, each rational number will only appear once and in its simplified form. While doing this, you will see that each fraction that appears on either side of 1/1 will have its reciprocal on the opposite side. This set is also an example of aleph-null.

                  Gardner goes on to other examples such as the card and 3 objects method which he says could go over aleph-null but dials it down to make it countable in Figure 25.1. What he does is say every card that’s the same color on a diagonal is turned over. But this still does not create a subset that can be on the list because its nth card is not the same as the nth card of subset n. The set of all subsets of an aleph-null is a set with 2 raised to the power of aleph-null. This is proof that a set like this cannot be matched one-to-one with counting integers because it is an uncountable infinity/ a higher aleph.

                  Cantor also proved that the set of real numbers is uncountable as well. It is described as given that there is a line segment going from 0 to 1. Rational fractions are in between and obviously in between the rational points being an infinity of other rational points. Following the card demonstration and every facing up card is replaced by 1 and every face down card is a 0. When you put a binary point in front of each row, there will be an infinite list of binary fractions between 0 and 1. Cantor proved that the three sets- the subsets of aleph-null, the real numbers, and the points on the line segment have the same number of elements. He called this C or “the power of the continuum”. He also called it aleph-one, the first infinity greater than aleph-null.

                  The distinction between aleph-null and aleph-one is as follows: aleph-null is countable natural numbers while aleph-one is countable ordinal numbers. The distinction is important for geometry because if an infinite plane is encounters that is tessellated with a polygon, the number of vertices would be aleph-null. When using ESP symbols, all but one can be drawn aleph-one times on a piece of paper.

The plus symbol cannot be aleph-one replicated.

                  Physicist, Richard Schlegal, tried to relate both alephs to cosmology by contradicting the steady-state theory (the universe has always existed and has always been expanding with hydrogen being continuously created [Google]).

                  One of the articles in the bibliography was “Non-Cantorian Set Theory”. The article gives an example about the counting numbers and the prime numbers like Gardner included on page 328 but the authors also included the fractions. I’m glad Gardner did not include their fractions because it makes the whole concepts more complicated than it needs to be. The authors touched upon the idea that there are an infinite set between aleph-null and aleph-one. The authors went more in depth with this concept and Gardner did include this in chapter 25 and I’m glad he did so that the whole chapter wasn’t so one sided towards a theory that has not been fully proved yet.

                  Gardner also included the Farey Sequences which is the explanation for the fraction and their reciprocals set. He grabbed some of these ideas from “Farey Sequences” from the Enrichment Mathematics for High School.

                  Lastly, Gardner used theories described in “The Problem of Infinite Matter in Steady- State Cosmology” by Richard Schlegal. Schlegal describes how the number of atom spaces begins with aleph-null and exponentially increasing. Gardner includes this idea in chapter 25 and goes on to say that the as the universe increases then so does the number of infinite alephs. I’m glad he included this because it makes the idea of increasing infinite numbers easier to understand by connecting it to the always increasing universe which we presume to be infinite as well. I wish he included the equation for this idea such as Schlegal did on page 21. He represented it by N = No e¯bt. Gardner did not attempt to describe this equation even though it goes hand-in-hand with the whole concept of expansion where N equals the volume of atom spaces (aleph-null) and b is constant creation/expansion.

                  I thought that this chapter was hard to understand at first but after reading it a few times, the idea of a number being larger than infinity became easier to perceive with the connection to sets and statistics as well.