• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 2.1. Countable Infinity
• 2.2. Uncountable Infinity
• 2.3. The Principle of Induction
• 2.4. The Real Number System
• 2.5. Pi and e are irrational
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

2.2. Uncountable Infinity

Example 2.2.9: The Continuum Hypothesis

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

• card(P(N)) = card(R)

In fact, this is a deep question called the continuum hypothesis. This question results in serious problems:

• 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.

Hence, it seems impossible to decide this question with our usual methods of proving theorems.

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 ?

Interactive Real Analysis, ver. 2.0.1(c)
(c) 1994-2018, Bert G. Wachsmuth
Page last modified: Jan 26, 2023