A friend of mine was trying to find a field with cardinality bigger than $\aleph_1$. Then I come up with a method that constructs the real number system by using arithmetical ultrafilters on $\mathbb{N}$, and we get $^*\mathbb{N}$, $^*\mathbb{Z}$, and $^*\mathbb{Q}$, where $^*\mathbb{Q}$ is a field. Then we pick elements in $^*\mathbb{Q}$ that follows Archimedes property to form $\mathbb{Q}_<$, and an equivalence relation $\sim $ on $\mathbb{Q}_<$ to form $\mathbb{R} = \mathbb{Q}_</\sim$.
It has been proved that the space of all ultrafilters on $\mathbb{N}$ is a huge space, with cardinality $2^{2^\omega}$. Then what is the cardinality of $^*\mathbb{Q}$? Also, does $|^*\mathbb{N}| = |^*\mathbb{Z}| = |^*\mathbb{Q}|$? If not, what are cardinalities of them?
While I'm not very sure about the cardinality of $^*\mathbb{Q}$, I can tell you that $|^*\mathbb{Q}|=|^*\mathbb{Z}|=|^*\mathbb{N}|$ mainly because $|\mathbb{Q}|=|\mathbb{Z}|=|\mathbb{N}|$, and hence their extensions will also have a similar cardinality.