God made nothing and nothing made everything

The great mathematician Kronecker apparently said “God made the integers, all else is the work if Man”. Since then mathematician have managed to build the integers from nothing. Well not quite nothing, but from the empty set, a set containing nothing, and a few rules. These rules prove enough to generate integers, rational fractions, real numbers, infinitesimals and infinite Integers. Fascinating though these are they are not, event he infinite integers, the main concern here. The main concern is what might be called “Classical” infinity, the theory largely developed by Cantor and which (probably incorrectly) is alleged to have driven him insane and to the study of Theology.

The lowest order of infinity

Start with the natural numbers

I= {1,2,3……..} going from one upwards. For every number there is a next number.

Now include the negative numbers and for completeness add Zero. Both negative numbers and zero were oppose by mathematicians of the day and are now accepted, though the notation for negative numbers, and even fractions has changed over the centuries. We get the integers proceeding from zero in each direction.

N = {………-2,-1,0,1,2,3…….}

Comparing infinities

Are there more integers than natural numbers? One of Cantor’s insights was that two sets are equal if the elements of one can be paired off with the elements of the other. Using –> to indicate pairing

0 –>1

1–> -1

2–> 2

3–> -2

and so on. There are as many integers as natural numbers.

How about the even numbers? We can pair each even number with half its value. 2 pairs with one and 4 with two. Zero, being contrary as always, pairs with itself (0/2 =0 after all).

A bigger infinity

Rational fractions can also be paired off with the integers. When we get to real numbers things get interesting. The real numbers, which include the irrationals and the aristocratic crowd of transcendental numbers, cannot be paired off with the integers. The proof is delightfully easy, Assume you have paired off all the real numbers with the integers, thus listing them all, and then construct a real number unpaired with any integer. Turn any real number into a number between 0 and one by putting a decimal point in front of it Making this mathematically precise would take too long). Now take the first number, paired with one, and start your new number with a different digit. Then take the second number and add a digit different from its second digit, and so on, You will end with a number that differs from every listed number in one place. This is called a diagonal argument.

If we say A0 is the infinity given by the size of the set of Integers, we have a second infinity given by the size of the set of real numbers. It does not stop there. It is possible to extend the set of real numbers between 0 and 1 to include the hyper-reals, smaller than any real number but not zero, and then the hyper-hyper reals.. Infinity seems to be breeding.

Infinite types of infinity?

Go back to the integers. It is relatively show the set of all subsets of a set ( known as its power set) is larger than the set even if the set is infinite. We can use this principle to create an infinite number of infinities each larger than the other. But are these the only possible infinities? The answer is undecideable.

Cantor named the infinity that corresponds to the size of the set of real numbers “c”. Mathematical typesetting is not possible there so we use P(X) for the power set of a set X and |X| for its size . Cantor’s continuum hypothesis is that |P(N)|=c that is that every real number corresponds to a subset of the integers. In 1963 this was finally shown to be undecidable though Gödel and Cohen, who between them completed the proof over a period of more than 20 years (but did not formally work together) apparently believed that if the theory of sets is properly extended the Continuum hypothesis will turn out to be false in the extended theory.

The Continuum hypothesis basically says c, the power of the continuum, is the next infinity after the integers. But there may be an arbitrary or even infinite number of integers between the two. No one has yet managed to find even an upper limit for what c might be. P(N) could be larger or smaller than c and even if the continuum hypothesis were true there may be other infinities not reachable by constructing power sets starting with the integers.

Speculation

Gödel believed mathematics describes reality, which is where we started and notes that if the Continuum hypothesis is undecidable then the description of reality is incomplete for in the real world the hypothesis must be true or false. In other words Gödel seems to have believed number has a reality outside the human mind. Investigating that assumption leads into the field of archetypes and Jungian Psychology and is a quest for another day..