We are back to Cantor for a quick discussion.

Recall that Cantor proved that

*card(N) < card(P(N)) < card(P(P(N))) < card(P(P(P(N)))) < …….*

Where does card(R) (the cardinality of the reals) fit into this the above sequence of inequalities?

As an application of the Cantor-Schroder-Bernstein theorem (card(A)≤card(B) and card(B)≤card(A) implies card(A) = card(B)) one can prove that

*card(P(N)) = card(R*)

This leads us to the following question: Is there some subset X of the real numbers for which

card(N) < card(X) < card(P(N)) = card(R) ?

**Conjecture (Cantor 1877):** There is **no** set whose cardinality is strictly between that of the integers and that of the real numbers.

This conjecture is called the *continuum hypothesis.*

Hilbert posed this problem as one of his Paris problems in 1900.

K. Goedel (1940) showed that the CH cannot be disproved with ZFC (Zermello-Frankel-Choice)

P. Cohen (1963) showed that the CH cannot be proved with ZFC (Zermello-Frankel-Choice)

As it turns out, the CH is true for “most” practical sets.

**Theorem** (P. Aleksandrov 1916): The CH is true for all Borel subsets of the real line.

We point the student to this link for a further discussion on this.

### Like this:

Like Loading...