2.2. Uncountable Infinity

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

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