Ordinals: The roof of finite numbers

Ordinals, the topic of our discussion, are immense. Get ready as we explore the limitless potential of ordinals! We'll begin by examining omega!

Omega

Omega is described as the largest finite number, which you can count to, although it would take an eternity. Infinity differs from this. Infinity is such that, no matter how long you count, you would be no nearer to reaching it than when you started. Conversely, Omega is less than Omega+1—there is a distinction, but now it's time to grow larger!

Epsilon-0 and Epsilon-Epsilon 0

Epsilon-0, or E0 for short, is an expansion of Omega. Remember when we said O+1>O? Can't we do O+2, O+3......O+O? Of course we can! Now, we can do this until we get 3O, then 4O.....O^2! Now, by increasing the power, we get O^3.........O^O! Than O^O^O! Than O^O^O^O^O.........O times! Do you want to write it all out? I'm presuming not. But that is E-0!

Wait a moment. Doesn't O^O.....O times remind you of something from another post? Yes, it's the well-known Tetration! In short, O^^O equals E0! But why does E0 have a zero? Can we modify that? Absolutely! E1 is E0^E0.....E0 times, which is E0^^E0! We can continue, E1^^E1 until we reach E(E0)! If we express this in terms of E0, we get ((E0^^E0)^^(E0^^E0)^^.... Which is pentation! In summary, E(E0) is E0^^^E0!

Zeta-0: A quick view

Remember when we stated E(E0)=E0^^^E0? Why can’t we have E(E(E0), or E(E(E(E0)? Of course we can! In fact, there is a new ordinal to repersent E(E(E………)))))), and it’s Zeta-0! Because of this new infinite nesting of Epsilons, and every time we pentate E(E(0) there is one more E, and we need to pentate epsilon times, we can write this: E0^^^^E0=Zeta-0!