L'infinito...ahhhh...ahhhhh

There was another deep problem stemming from the budding theory of cardinal numbers that occupied Cantor and which was not resolved during his lifetime. Because of the importance of countable sets, the symbol ℵ₀ (“aleph zero”) is frequently used for card N. The subscript “0” is appropriate when we remember that countable sets are the smallest type of infinite set. In terms of cardinal numbers, if card X < ℵ₀, then X is finite. Thus, ℵ₀ is the smallest infinite cardinal number. The cardinality of R is also significant enough to deserve the special designation c = card R = card (0, 1). The question that plagued Cantor was whether there were any cardinal numbers strictly in between these two. Put another way, does there exist a set A ⊆ R with card N < card A < card R? Cantor was of the opinion that no such set existed. In the ordering of cardinal numbers, he conjectured, c was the immediate successor of ℵ₀.
Cantor’s “continuum hypothesis,” as it came to be called, was one of the most famous mathematical challenges of the past century. Its unexpected resolution came in two parts. In 1940, the German logician and mathematician Kurt Gödel demonstrated that, using only the agreed-upon set of axioms of set theory, there was no way to disprove the continuum hypothesis. In 1963, Paul Cohen successfully showed that, under the same rules, it was also impossible to prove this conjecture. Taken together, what these two discoveries imply is that the continuum hypothesis is undecidable. It can be accepted or rejected as a statement about the nature of infinite sets, and in neither case will any logical contradictions arise.
The mention of Kurt Gödel brings to mind a final comment about the significance of Cantor’s work. 
Gödel is best known for his “Incompleteness Theorems,” which pertain to the strength of axiomatic systems in general. What Gödel showed was that any consistent axiomatic system created to study arithmetic was necessarily destined to be “incomplete” in the sense that there would always be true statements that the system of axioms would be too weak to prove.