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Infinity is one of those ideas which is a heavily researched and interesting idea in mathematics which has very little applicability elsewhere. Check out an int

Infinity is one of those ideas which is a heavily researched and interesting idea in mathematics which has very little applicability elsewhere. Check out an intro analasys book for more details on countable and uncountable sets. The cantor set especially, is very interesting as a set with measure 0 and still uncountable many elements. PS. the use of aleph-0 for countable sets is discuraged as we do not know is there exists a set with cardinality between the countable and uncountable sets (ie. a set which does not have a bijection with the natural numbers but does have a bijection with a "small" subset of the reals such that it did not have a bijection with the reals themselves. ). Its called the continuum hypothesis. Details here: http://en.wikipedia.org/wiki/Continuum_hypothesis http://en.wikipedia.org/wiki/Continuum_hypothesis

joe_the_user 18y ago
Actually, the theories about infinity are pretty much necessary for calculus, which is pretty much necessary for most physics, especially practical physics. So the basic properties infinity is, *indirectly, a very practical and applicable topic. Also, The continuum hypothesis is not a statement of "unknown" truth like the Rieman hypothesis but rather a statement which been proven independent (or "unprovable) from the axioms of ordinary (Zermelo-Franklin) set theory. It is not that we don't know whether there exist sets of a cardinality between that of the integers and that of the reals. Rather, we know there exist consistent models of set theory where the statement is true AND consistent model where it is false. I also haven't heard of the alpha-null notation being discouraged either - it is used in the article you link-to.
gjm11 18y ago
The use of aleph-0 for countable sets is not discouraged. I think what you mean is that using aleph-1 to mean "of the same cardinality as the reals" is discouraged, because it's only appropriate in models where the continuum hypothesis holds. There's a not-so-commonly-used notation using the Hebrew letter beth instead of alpha, where beth_0 = aleph_0 and beth_{n+1} = 2^beth_n (note for experts: and you do the obvious thing at limit ordinals). So the cardinality of the real numbers is beth-1, whatever model you're in.

joe_the_user 18y ago

Actually, the theories about infinity are pretty much necessary for calculus, which is pretty much necessary for most physics, especially practical physics. So the basic properties infinity is, *indirectly, a very practical and applicable topic. Also, The continuum hypothesis is not a statement of "unknown" truth like the Rieman hypothesis but rather a statement which been proven independent (or "unprovable) from the axioms of ordinary (Zermelo-Franklin) set theory. It is not that we don't know whether there exist sets of a cardinality between that of the integers and that of the reals. Rather, we know there exist consistent models of set theory where the statement is true AND consistent model where it is false. I also haven't heard of the alpha-null notation being discouraged either - it is used in the article you link-to.

gjm11 18y ago

The use of aleph-0 for countable sets is not discouraged. I think what you mean is that using aleph-1 to mean "of the same cardinality as the reals" is discouraged, because it's only appropriate in models where the continuum hypothesis holds. There's a not-so-commonly-used notation using the Hebrew letter beth instead of alpha, where beth_0 = aleph_0 and beth_{n+1} = 2^beth_n (note for experts: and you do the obvious thing at limit ordinals). So the cardinality of the real numbers is beth-1, whatever model you're in.