Infinity and beyond: a simple view of the continuuum hypothesis

Author:
AlexK2009
Aug 15, 2015

Infinity is a subject of (naturally) endless interest. Mathematical infinity, spatial infinity, infinite time, and the paradoxes involved have interested ( and perhaps deranged) some of the most powerful minds on this planet.

And infinity keeps cropping up in the strangest places, for example in the foundations of mathematics or, in Physics, generally as a sign that a theory is being used where it begins to break down.

With the renewed interest in parallel universes we now also have the possibility of an infinite number of universes and an infinite number of copies of each of us.

Lets stick to numbers. Then go to infinity and beyond.

Numbers

Like time everyone knows what a number is- until asked to say what it is. Mathematicians share this problem with every one else but know how to build numbers, many ways to build numbers. Perhaps the simplest is to start with sets, the building blocks of mathematics.

The simplest definition of a set is due to Georg Cantor, who said a set is a multiplicity that can be considered as a unity, for example a herd of sheep, a flock of geese, a corruption of politicians ( or in the case of world leaders a lack of principals) or a murmuration of starlings. Anything in a set is an element of a set and a set, if written out in full, is written as a pair of curly brackets enclosing its elements. Of course it would be hard to fit a herd of sheep inside curly brackets (they fit better inside a pen - a sentence that is a well known trap for AI and translation software), so normally just the names of the elements are used.

Start with the empty set written as ∅ = {} which is not nothing but contains nothing. Since Physicists believe that at the lowest level particles are constantly being created from nothing and popping back into nothing and bankers create money from nothing and then pop it back into nothing leaving a recession and a load of rich bankers, it is clear that nothing is important. And like an empty bank account the empty set is potentially quite productive.

So we define zero as the empty set. Then we define 1 as the set containing the empty set 1 = {∅} and two as {0,1} and so on. In this way we can get all the so called natural numbers or positive integers. More complicated ways of creating these numbers are possible and they also allow negative numbers and fractions, but these are out of scope here.

Clearly this process never ends, leading to a loose definition of infinity as “always one more”. We cannot list all the integers but we can imagine the set of all integers which we write as N

and we use A_o to for the size of this set. Mathematicians normally use strange symbols and, following Georg Cantor, Hebrew letters, for infinitely large numbers, but these are painful to render on a web page.

The smallest infinity

A_ois the smallest known infinity and has the interesting property

A_o+ A_o= A_o

Ao + 1 = A_o

If you have an infinite number A_oof pairs of shoes and add one pair it is still infinite, A_o

If you take separate the left and right shoes of each pair you have two heaps of equal size and an infinite number of happy one legged beggars. Hence A_o+ A_o= A_o

This becomes easier to see using Cantor's insight that two sets are equal if every element of one can be paired with every element of the other with nothing left over. More formally this is called a one-to-one correspondence. An infinite set is one that can be put into a one to on correspondence with a subset of itself.

The standard example of this is the set of even numbers {2,4,6,8.....} which can be put into a one to one correspondence with each natural number 1-->2, 2--->4 etc.

It can be shown that the set of all fractions m/n where m and n are natural numbers is the same size, A_oas the set of natural numbers.

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Any set that can be put into correspondence with the set of natural numbers has size (cardinality) A_o and is called countable.

Larger infinities

Suppose we have an infinitely large library with all books that can be written using the letters of the alphabet, the digits 0 to 9 and a space. So one book is empty, and there is a book with an infinite number of “a”s and so on. Somewhere in this lot is the answer life, the universe and everything and also the question to which this relates. But that is for another day. Start by giving each book a unique number 1,2,3 etc, in effect creating a catalogue of books. There are A_obooks, obviously.

Now construct a new book not in the library. Start with the first character of the first book and write down a different character. Then look at the second character of the second book and write down a different character and so on. In the end you will have a book that differs from each book in the library by at least one character.

The paradox here is assuming the library holds all possible books and constructing a book that is not in the library, which violates the assumption that the library holds all possible books. What we actually did is take a countably infinite set of books and prove that some books in the library are not in that countable set This can only be true if the library is bigger than A_o.

To look at it another way this sneaks in Cantor's proof the there are more real numbers than integers, for the content of each book can be transformed into a unique number between 0 and 1 and then we have a pairing of the integers with real numbers, but can construct a number that is not paired with any integer. The books are discrete objects, much like integers but are uncountable.

This is why some people think the study of infinity is dangerous.

The continuum hypothesis

This is what may have driven Cantor insane and made his spiritual successor Kurt Godel paranoid, though others have followed them into this maze and returned sane. We need to start with the power set of a set, the set of all subsets of a set (including the empty set and the set itself). For a set with n elements there are 2ⁿ subsets.

We saw from the infinite library that the collection of books cannot be listed or indexed by the integers, it is uncountable. It is known that

2^Ao= c

where c is the (infinite, obviously) number of points in a finite length perfect line. And this is the number of real numbers between zero and one. So is the number of books in the library c or not?

The continuum hypothesis is that there is no infinity between these two. Implying there are c books in the library.

Now if the continuum hypothesis were true then the sequence of infinities would go on for ever with each new one being the size of the power set corresponding to the previous set. More formally any infinity k must be 2^An for some n, suggesting a countable number of infinities. Another way of putting this is that for all n then there are no infinities between 2^Anand 2^An+1. it would also mean each infinite set other than A_o is the power set of a smaller infinite set.

If the Continuum Hypothesis were false there could be an infinite number of infinities between each of these “regular” infinities. It also implies some infinite sets are not the power set of a smaller infinite set.

Unfortunately the Continuum hypothesis is not decidable given the starting assumptions (axioms) of set theory. This is a gaping hole in the floor of mathematics. It could have meant constructing two different and consistent mathematics depending on whether you assume the hypothesis to be true or false.

Mathematicians like this as much as religions like the idea that other religions with different assumptions are equally valid or politicians like the idea that their opponents may have got something right. Fortunately mathematicians, perhaps uniquely among humans, have a non violent way to settle their differences. Simply tack an “obviously” true assumption on to the assumptions already used in set theory (This is theory. In practice until a proof is provided mathematicians are as fractious as any other group, sports fans for example).

And there is the rub. What assumption to use? This is an active area of research among mathematicians who are willing to risk their sanity and most of the work is currently inaccessible to anyone outside the field. Perhaps anyone who takes up this research should be aware it comes with a risk to sanity clause. But, as Groucho Marx allegedly said “There ain't no sanity clause”