On Continuum Hypothesis

** There is no relationship between Continuum hypothesis, whether true or false, and Riemann hypothesis (as it were told by someone in his message to me on Twitter, and of course, we wouldn’t have anything to construct such intuition). Riemann hypotesis deals with all nontrivial zeros of the Riemann zeta function on critical line x=1/2. Riemann hypothesis (that is known as Hilbert’s eight problem) is still unresolved and is considered to be the most important outstanding problem in contemporary number theory, especially in the distribution of prime numbers. Incidentally, Continuum hypothesis is also regarded by many people as the most important hypothesis in Set Theory. They both share the most important hypothesis in their areas (and presumably, Riemann hypothesis implies important result in many areas, if it is valid, but of course we are still in the realm of conjectures and speculation).**

Continuum Hypothesis deals with ‘something’ in between set A and the set of subsets of A. As I wrote in my previous posting, there is an infinite hierarchy of infinite sets. Cantor then raised the following question : If $A$ is the set of positive integers, we know that $A < P(A)$, but is there some set $B$ such that $A < B$ and $B < P(A)$? Cantor conjectured that the answer is ‘no’, and it is this conjecture that is called the Continuum hypothesis. (You might be interested in advanced reading about this topic in Infinite Ink. Update : Interesting paper by W. Hugh Woodin on how ZFC might be modified).

It worths to briefly point out some interesting ideas underlying the Continuum hypothesis. Continuum hypothesis is called by many people as the parallel postulate of set Theory. It has been proved by Kurt Godel and Paul Cohen, that neither the continuum hypothesis nor its negation follows from the ‘basic’ axioms of set theory, and no one has yet been able to produce a not-so-basic axiom that would yield a convincing answer to Cantor’s question.

A similar thing happened in the case of the well ordering principle, an axiom equivalent to the axiom of choice. Cantor then adopted it. The Medievals had noted that the number of points in a large circle is the same as that in a small concentric circle, in the sense that each radius of the large circle passes through exactly one point of each circle. Similar observation led Bernhard Bolzano (1781-1848) and others to the conclusion that any two infinite sets are ‘equal’ because they can be linked by a one-to-one correspondence. In 1873 Cantor discovered that this is wrong. One of his proofs goes as follows.

Let $A$ be an infinite set (that is, one containing infinitely many ‘numbers’). Let $P(A)$ be the set of subsets of A. Suppose that $A$ and $P(A)$ are linked by one-to-one correspondence $f : A \rightarrow P(A)$. Let $S$ be the set of members $x$ of $A$ such that $x \not\in f(x)$. Then $S \in P(A)$, and there is some $y \in A$ such that $S = f(y)$.

If $y \in S = f(y)$ then, by defining property of $S$, $y \not\in f(y).$ However, if $y \not\in f(y) = S$, then, by the definition of $S$, it is a member of $S$. Contradiction. Hence $A$ and $P(A)$ are not linked by one-to-one correspondence.

Since for every member $x$ of $A$, $P(A)$ has $\{x\}$ as a member, there is a ‘copy’ of $A$ that is a subset of $P(A)$. Hence $A$ is smaller than $P(A)$, and we can write $A. Similarly, $P(A). And indeed, we arrive at the infinite hierarchy of infinite sets (as you see at my previous posting), each more infinite than the previous ones.