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Was asked this question, and I have no idea about how to start proving it.

Could someone give me some good reference material to start with.

yiyi
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    It is unproveable in ZFC set theory if there is a cardinal between $\aleph_0$ (rationals) and $2^{\aleph_0}$ (irrational). – William Sep 17 '12 at 06:21
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    This is called the continuum hypothesis. Googling will give you good results. – Alex Becker Sep 17 '12 at 06:21
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    Dear William: it is quite unlikey that you comment is of help to the OP... – Mariano Suárez-Álvarez Sep 17 '12 at 06:22
  • @MarianoSuárez-Alvarez at least it give me terms that I should look up and learn more about. Nothing is really lost in asking stupid questions, even if others answers are for now beyond my knowledge. – yiyi Sep 17 '12 at 06:23
  • To elaborate. The question is related to the continuum hypothesis. You can produce models of $ZFC$ where $\aleph_1 = 2^{\aleph_0}$ and a model where $\aleph_1 < 2^{\aleph_0}$. In the former, there is no cardinality in between. In the latter, there is. Hence the axioms of $ZFC$ can not prove the continuum hypothesis. – William Sep 17 '12 at 06:24
  • @William this is the cool math that should be taught earlier in ones schooling. – yiyi Sep 17 '12 at 06:26
  • I agree. The concept of bijections and cardinality can be introduced fairly quickly. However the independence result is often done using constructibility and forcing, which are more technical. – William Sep 17 '12 at 06:29

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The assertion that there is none is called The Continuum Hypothesis, and it is provably unprovable from the usual axioms of mathematics. Cantor made this hypothesis some 150 years ago, and it was in fact the first on Hilbert's problems.

Historically Cantor asked this a long time before there was a formal foundation to set theory. Some time into the 20th century the formulation of set theory as we know it today began to take shape. Now we use an axiom system called ZFC, originally formulated by Zermelo, with additional axioms by Fraenkel, and with the axiom of Choice adjoined.

Gödel proved in the subsequent years that if a theory is consistent then it has a model, and much later he showed that if ZF is consistent then ZFC+CH (and more) is consistent. He did this by constructing what is known as The Constructible Universe in modern set theory.

In 1963 Paul Cohen came up with a technique which is known as Forcing, in which we begin with a model of set theory and we extend it by adding a particular set which has certain properties. Using this technique he showed that if ZFC+CH is consistent then it is so is ZFC+"CH is false".

Combining the two results we have that from the usual axioms of set theory (namely, ZFC) we cannot prove nor disprove the continuum hypothesis. Both Gödel's work as well Cohen's are complicated and technical. Most books about axiomatic set theory should cover them at some point. For example Kunen's or Jech's.

Asaf Karagila
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