${\aleph }_1$ is the next large cardinality (after ${\aleph }_0$) and you can show that such a cardinality and successive ones can exist. A set with a cardinality larger than ${\aleph }_0$ is called uncountably infinite. It was (is?) suspected that the cardinality of the set of reals (and also the set of irrationals) is ${\aleph }_1$ but it has been shown by Godel and Cohen that this cannot be disproved or proved (respectively)

Marcos

P.S. In case you're not sure what cardinality means: for a finite set, it's the number of elements in the set. You can (sort of) visualise what happens intuitively with infinite sets, but I won't go into more detail on cardinalities of infinite sets unless you want me to.

1) why ${\aleph }_1$ is the cardinality of the set of real, not the cardinality of the set of complex?

2) What are the defintions of ${\aleph }_3$, ${\aleph }_4$, ... ${\aleph }_n$? (I know ${\aleph }_2$ is the cardinality of the set of complex number)

3) Why ${\aleph }_1$ is the power set of ${\aleph }_0$? (same question as Marcos' one)

4) Why ${\aleph }_{n+1}>{\aleph }_n$? Is there a general proof? Is it still true when $n$ tends to infinity?

KL Kam

P.S. $\aleph$ (aleph) is a Hebrew alphabet.

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Quote:

What Godel and Cohen actually proved is that the continuum hypothesis is independent of the Zermelo-Fraenkel axiom system...

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Yatir, I know but phrasing it as "this can't be proved or disproved" sounds more spectacular as it sounds like a completely contradictory statement (Also, I wasn't sure if the person reading the post was familiar with the axioms of set theory)

Marcos

Consider the following table (where the rational in the mth column, nth row is n/m):

Hence, cardinality is ${\aleph }_0$

Marcos

P.S. You can quite easily use this very simple technique to show that $N^2$ has cardinality ${\aleph }_0$ by changing $n/m$ into $(n,m)$ (without going into other more subtle proofs)

Demetres