1
$\begingroup$

Are there any examples of an algebra on elements belonging to a set of greater cardinality than that of real numbers. Can a reference to their use be given?

$\endgroup$
  • $\begingroup$ continuous functions on stone-cech compactification of $\mathbb{N}$. $\endgroup$ – Norbert Nov 11 '13 at 7:22
  • $\begingroup$ @Norbert Thanks. This would be an example of domain having the cardinality $\aleph_2 / 2^\mathfrak c$ are their examples of even greater cardinalities. $\endgroup$ – ARi Nov 11 '13 at 7:40
1
$\begingroup$

It is consistent that the real numbers have cardinality $\aleph_2$, in which case a function $f\colon\Bbb{R\to R}$ satisfies your requirements.

If you want a set whose cardinality is explicitly $\aleph_2$ then you will have to resort, in one way or another, to $\omega_2$ which is the second uncountable ordinal (i.e. the cardinal $\aleph_2$). The function can be anything, from the successor ordinal, to anything crazy.

However, I feel that you're not talking about $\aleph_2$, but rather $\beth_2$ i.e. $|\mathcal{P(P(}\Bbb R))|$. In that case you can consider the Lebesgue measure as a function whose domain has cardinality $\beth_2$, and its range is $\Bbb R$.

(Also related: Functions on P(R) - are there examples?)

$\endgroup$
  • $\begingroup$ "It is consistent that the real numbers have cardinality $\aleph_2$...isnt $\aleph_1$ the cardinality of the set of real numbers? $\endgroup$ – ARi Nov 11 '13 at 8:37
  • $\begingroup$ math.stackexchange.com/a/490420/622 There are probably many other questions and answers about this question. But no. The cardinality of the real numbers is not necessarily $\aleph_1$. $\endgroup$ – Asaf Karagila Nov 11 '13 at 8:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.