Are the following sets fields: the empty set, {0} {0,1}? (it's that I've seen {0,1} as an example of a field yet I thought for any element of a field, there must be another element such as the sum of the two is equal to zero. Also while I'm asking silly questions: what is the cardinality of the hyperreals?
a field has to have at least two elements, so {0,1} is the smallest field. 1+1=0. the hyperreals are carved out of sequences of real numbers in one approach. the number of sequences of real numbers is aleph_2, i think. but i'm not sure how much of aleph_2 is carved out. card(R*) is either aleph_2 or aleph_1=card(R).
I'm unfamiliar with the hyperreals, but the set of all sequences of real numbers has cardinality C since a hilbert-hotel type apprach will create a bijection between sequences of real numbers, and individual real numbers.
The fundamental difference between a set and a field is that a set (by itself) has no binary operations. A field is a set with two operations (and inverses) satisfying a whole collection of rules. The operations are generalizations of addition and multiplication. The cardinality of the reals is usually designated by C (continuum). The continuum hypothesis states that C=aleph_{1}. Under the generalized continuum hypothesis, the set of all subsets of the reals has cardinality aleph_{2}.
As a matter of taste, it's probably better to say [tex] |{}^*\mathbb{R}| = 2^c [/tex] So you don't have to talk about the continuum hypothesis. The construction of the hyperreals goes as follows: We have a magical thing, called an ultrafilter, which tells us whether a subset of N is "big" or "small". It has the properties that if A is a big set, then the complement of A is a small set. It also has the properties that all finite sets are small sets, and if A is big and B contains A, then B is big, and the union of two small sets is small. (I think you need the axiom of choice to prove ultrafilters exist) Using this ultrafilter, we can define an ordering relation on sequences of real numbers. If s and t are sequences of real numbers, then: [tex] s < t~\mathrm{if~and~only~if}~\{n \epsilon \mathbb{N} | s_n < t_n\}~\mathrm{is~big} [/tex] And similarly for any other ordering operation (including equality).
The point of the generalized continuum hypothesis (gch) is 2^{c}=aleph_{2}. Without gch, the equation is unprovable.