**Aleph-Null and Aleph-One**

Martin starts out by mentioning a question raised by Paul J. Cohen (29) in 1963. Is there an order of infinity higher than the number of integers but lower than the number of points on a line? To answer this, Martin goes into an explanation of Georg Cantor’s discovery of the infinity of higher levels of infinity. The base level, the number of integers, was called Aleph Null. We include in this set any selection of numbers that can be counted (not that anyone is actually going to try counting them), such as all prime numbers, all natural numbers, all even, odd, positive or negative numbers, all rational fractions, and so on. Yes, some of these sets can include values that get much bigger, much faster than some of the other sets, but the important distinction is that they can be counted.

Example with prime numbers:

1 (1), 2 (2), 3 (3), 5 (4), 7 (5), 11 (6), 13 (7), 17 (8), 19 (9), etc.

Aleph-Null numbers are said to be “countable” or “denumerable”. This brings us to a paradox, in that with infinite numbers, “they can be put into correspondence with a subset of themselves,” or with their “subsets.” One example was developed by logician Charles Sanders Pierce. Start with the fractions 0/1 and 1/0 (ignoring the divide by zero issue for the moment). Sum the two numerators and then the two denominators to get the new fraction 1/1 and place it between the original pair. Repeat this process with each pair of adjacent fractions to get the new fractions that go between them.

0/1 . 1/0

0/1 . 1/1 . 1/0

0/1 . 1/2 . 1/1 . 2/1 . 1/0

0/1 . 1/3 . 1/2 . 2/3 . 1/1 . 3/2 . 2/1 . 3/1 . 1/0

As Gardner mentions, as this series continues, every rational number will appear once and only once. Reducible fractions like 10/20 never show up. If you want to count the fractions at any given step, you can take them in their order of appearance. Any two fractions equally distant from the center, 1/1, are reciprocals of each other. And, any two adjacent pairs ab and c/d have the following equalities: bc – ad = 1; c/d – a/b = 1/bd. Also, as Gardner says, this series is closely related to Farey numbers. Pierce’s process creates an infinite number of countable fractions, as a subset of all numbers.

(All rights belong to their owners. Images used here for review purposes only. Example of creating subsets of a set of three elements.)

The next step is to show that there’s another set with a higher order of infinity than Aleph-Null. The above figure uses three objects – a watch, a key and a ring. Below each, you lay out a row of 3 cards, face down (black) or face up (white) to represent subsets. A white card shows that the above object is in the subset, while black indicates that it is not. The first subset consists of the original set. The next three rows are subsets of 2 objects each, the following three are subsets of one object each, and the bottom row is the empty, or null, subset that doesn’t have any objects. “For any set of n elements, the number of subsets is 2^n.”

Apply this procedure to an Aleph-Null set, which is countable, but infinite. The above figure attempts to show that this new set is countable, with each row represented by the white and black cards extending to infinity. They can be listed in any order, and numbered 1, 2, 3, etc. “If we continue forming such rows, will the list eventually catch all the subsets? No – because there is an infinite number of ways to produce a subset that cannot be on the list.” We can do this by taking every card on the diagonal line and flip it white-for-black, making a new subset that hadn’t existed before, even though the original set was infinite. This set is not countable, showing that the assumption was wrong. So, “the set of all subsets of an Aleph-Null set is a set with the cardinal number 2 raised to the power of Aleph-Null.” As a side effect, Cantor’s diagonal proof shows that the set of real numbers (rationals and irrationals) is also uncountable. Again, using the above figure, we’ll assume a line segment with end points at 0 and 1. Assign binary values to the decimal portion of the segment, and list them on the rows of subsets, and you get the same result as before.

Cantor then showed that the subsets of Aleph-Null, the real numbers and the total points on a line segment all have the same number of elements. He gave this cardinal number the name “C,” the “power of the continuum,” and said that this is Aleph-One, the first infinity greater than Aleph-Null.

Gardner continues to talk about the importance of the Cantor sets in geometry, using the above figure. Say you have an infinite plane tessellated with hexagons. “Is the total number of vertices Aleph-One or Aleph-Null?” The answer is – Aleph-Null, because you can count them along a spiral path. But, the number of circles of one-inch radius placed on a sheet of paper is Aleph-One, because inside any small square near the center of the page are an Aleph-One number of points that are the centers of their own 1-inch circles.

**Puzzle for today:**

Using J. B. Rhine’s ESP card symbols, are there any symbols that can be drawn an Aleph-One number of times on a sheet of paper, assuming lines of no thickness, with no overlapping or intersections of the lines? (They don’t have to be the same size, but they must have similar shapes.)

To come back to Cohen… Cantor thought there would be an infinite hierarchy of Alephs, obtained by raising the previous one to the power of 2, and that there are no other Alephs in between. There’s no Ultimate Aleph, either. He tried, but could not prove that there’s no Aleph between Aleph-Null and C. Kurt Godel showed in 1938 that the so-called Cantor conjecture could be assumed to be true. And here we get Cohen, who proved in 1963 the opposite could also be assumed – that C is not Aleph-One. There can be at least one Aleph between Aleph-Null and C, although no one has any idea how to specify it. All he did was demonstrate that the continuum hypothesis is unprovable within standard set theory.