The Continuum hypothesis and the Generalized continuum hypothesis

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The previous section we mention continuum hypothesis and how angry Cantor became when König tried to prove it was false. In this section we will dig deeper into what the continuum hypothesis is about, and even consider a more generalized form – the generalized continuum hypothesis.

What is Continuum Hypothesis?

‘Continuum’ indicates sets of points that have certain continuity property. The informal definition of continuity commonly introduce in school is a real interval where it is possible to move from one point into another point in a smooth fashion without leaving the interval.

There are many sets that behave as a continuum:

1. the intervals or
2. the entire real line
3. the x-y plane
4. a volume in 3 dimensional space

The cardinality of the continuum is denoted as c.

Our concept of dimensionality leads us to think that things of higher dimension is larger (cardinality measure) than those of lower dimension. This preconception is false as we can see by demonstrating that a 1x1 square can be put in one-to-one correspondence with the unit interval. In notations we have the 1x1 square as and the we let I denote the open unit interval (0,1).

We can use the CBS theorem to prove that S and I are equinumerous - we just need to find injections from I to S and vice versa.

mapping from 1x1 square to a unit interval

Given an element q in I, we can map it injectively to the point (q,q) in S. We can also map in the reverse direction, consider a point (a,b) in S and write out the decimal expansions of a an b:

From the decimal expansions, let us create the decimal expansion of a number in I by interleaving the digits a and b. The image of (a,b) is given then as,

Cases such that two different points (x,y) get mapped to the same value s implies that each position of their decimal expansion must really be equal then we can say that they must be the same point. To prove that the function from S to I is bijective is a little bit harder.

We’ll start by presuming that all decimal expansions are non-terminating and use the following approach:

write out the decimal expansion of the coordinates of a point (a,b) in S.

For example, we have the following coordinate points:

From this non-terminating expansion we can form the digits into blocks with as many 0s as possible followed by a non-zero digit.

a being expanded into blocks	b being expanded into blocks

And the number is formed by interleaving them, for instance,

From here we can deduce that the unit square S, and the unit interval, I, have the same cardinality.
Now, let’s turn to the continuum hypothesis.

From previous chapter we talked about the cardinality of being denoted by (aleph nought). From this it seems obvious that there must be a higher cardinality, such as and . But what are these for?

In Cantor’s perspective on cardinality, he thinks that there was a sequence of cardinal numbers that give all of the possible infinities. The smallest among this sequence of set that Cantor can think of is the sequence of natural numbers, so Cantor called the cardinality . And accordingly, whatever the “next” infinite cardinal is, it is called the . It may seem inconceivable to produce the “next” infinite cardinal after , but we’ve encountered from Cantor’s theorem that there is a way to build some infinite cardinal bigger than using a power set construction.

The continuum hypothesis just says that this bigger cardinality that we get by applying the power set construction is that “next” cardinality we’ve been talking about.

The power set of the natural numbers is equivalent to the interval (0,1) which is one of the sets whose cardinality is c. The continuum hypothesis comes down to the following equivalence:

To this day, no one else knows the validity of the continuum hypothesis. It lives in a really weird world. The weirdness of this hypothesis includes the fact that it appears to be impossible to decide its validity. And the badass part of this is that it has been proved that one can’t prove the continuum hypothesis. In addition,

it has been proved that one can’t prove the continuum hypothesis.

You may have asked while wondering how the last 2 sentence were true: “with respect to what axioms?”.

For mathematics, the usual axioms are called the ZFC – Zermelo-Frankel set theory axioms together with the axiom of choice.

Paul Cohen determined that “CH is independent of ZFC.” More pedantically:

it is impossible to either prove or disprove the continuum hypothesis within the framework of the ZFC axiom system.

Generalized Continuum Hypothesis

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The main point of the GCH is that the only way to get from one infinite cardinality to the next is through power set construction. In other words, we can obtain every other aleph number by applying the power set construction a bunch of times.

^{Disclaimer: this is a summary of section 8.5 from the book A Gentle Introduction to the Art of Mathematics: by Joe Fields, the content apart from rephrasing is identical, most of the equations are screenshots of the book and the same examples are treated.}

1. A Gentle Introduction to the Art of Mathematics by Joe Field

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