Interactive Real Analysis
- part of Example 2.2.9: The Continuum Hypothesis
Is there a cardinal number c with card(N) < c < card(R) ?
What is the most obvious candidate ?
Back
We need to find a set whose cardinality is bigger than N and less that
that of R. The most obvious candidate would be the power set of N.
However, one can show that
Back- card(P(N)) = card(R)
- In the 1940's the German mathematician Goedel showed that if one denies the existence of an uncountable set whose cardinalities is less than the cardinality of the continuum, no logical contradictions to the axioms of set theory would arise.
- One the other hand, it was shown recently that the existence of an uncountable set with cardinality less than that of the continuum would also be consistent with the axioms of set theory.
Such undecidable questions do indeed exist for any reasonably complex logical system (such as set theory), and in fact one can even prove that such 'non-provable' statements must exist. To read more about this fascinating subject, look at the book Goedel's Proof or Goedel, Escher, Bach as mentioned in the reference section of the glossary.
Can you find sets with cardinality strictly bigger than that of the continuum ?