Uncountable Omega Fixed Point

Unlike a regular Omega Fixed Point which contains a countably infinite amount of omegas, an Uncountable Omega Fixed Point contains an uncountably infinite omegas (

$$\aleph_1$$ ). The existance of such an ordinal can be proven using the following steps: - For the regular Omega Fixed Point, label every omega with a consecutive natural number, starting from the first omega which will be labeled as "

$$a$$ ", then the second as "

$$a_1$$ ", the third as "

$$a_2$$ ", the fourth as "a3", etc.. The Omega Fixed Point will be notated as {

$$a$$ ,

$$a_1$$ ,

$$a_2$$ ,etc..}

- For the Uncountable Omega Fixed Point label every omega the same way, however we label each one with a consecutive REAL NUMBER. We start from the usual "a", but then we move on to a0.000000...001, and using the same proof that is used to prove there are more real numbers than natural numbers, we prove that there are more omegas in the Uncountable Omega Fixed Point than in the regular one, thus giving us a much, much bigger ordinal.