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Kind of. Alef-null is (I believe.) the infinite set of natural numbers. C is the infinite set of real numbers, called "continuity" because of its continuous property. It's an open problem of set theory whether there's any level of infinite set between the absolutely discrete alef-null and the absolutely continuous C.

Someone who's had more than a passing introduction to set theory could explain it better.

Also, if Alef is already a language, when will someone update it to make Bet?

Crazed123 wrote:Kind of. Alef-null is (I believe.) the infinite set of natural numbers. C is the infinite set of real numbers, called "continuity" because of its continuous property. It's an open problem of set theory whether there's any level of infinite set between the absolutely discrete alef-null and the absolutely continuous C.

Someone who's had more than a passing introduction to set theory could explain it better.

Also, if Alef is already a language, when will someone update it to make Bet?

Wikipedia, article Aleph=One:

Wikipedia wrote:Aleph-null (\aleph_0) is by definition the cardinality of the set of all natural numbers

Wikipedia, article Set:

Wikipedia wrote:\mathbb{C}, denoting the set of all complex numbers.

Candy: The set 'C' representing all complex numbers is denoted by a script capital C. As are Z,R,W etc for integer, real, rationals. This would differentiate it from another set, 'C'. (I'm sure that letter must have been used multiple times in set theory).

Aleph-null (\aleph_0) is by definition the cardinality of the set of all natural numbers

The cardinality of the set of all natural numbers - must mean that aleph-null is some form of infinity then... :S
[EDIT]

the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets.

I really should STFW before I post...
[/EDIT]

As crazed said, it is an open problem whether the cardinality of the set of all real numbers > that of the set of all natural numbers. Is one infinity bigger than another?

JamesM

I think this "C" means the cardinality of the set of real numbers, rather than the set of complex numbers. (the cardinality of C (superset of C) is C squared, equals C ).

And btw, C > Aleph-0. So much for an open problem.

Heh, they told me it was, in my set theory classes. Obviously they didn't want to complicate things by talking about aleph-null etc.

Combuster wrote:I think this "C" means the cardinality of the set of real numbers, rather than the set of complex numbers. (the cardinality of C (superset of C) is C squared, equals C ).

And btw, C > Aleph-0. So much for an open problem.

We know that C > Alef-null. The open problem is if they are the 1 and 0, respectively, of set cardinalities or whether any X exists for C > X > Alef-null.

Crazed123 wrote:We know that C > Alef-null. The open problem is if they are the 1 and 0, respectively, of set cardinalities or whether any X exists for C > X > Alef-null.

From what I've heard either of C = Aleph-1 and C != Aleph-1 can be used with current mathematics without losing consistency. To me, that sounds like its impossible to prove that only one is correct.

There must be some things in life we will never understand