You have missed the point. It is my mental model of the universe which consists of an infinite number of subsets, not the universe.
My mental model is a "well defined" mental construct which I created for the sole purpose of considering the consequences of my definition: "reality is a set of numbers. As my model must be capable of representing any possible concept of reality (i.e., any possible universe concievable ) if I set any limits on what is possible my deductions are simply wrong!
Since I must model any possible universe, that is any finite collections of "observations", my "model" must include the the possibility of another observation, no matter how many observations are examined. That is as a matter of fact, not opinion, the very definition of infinity!! Please read more carefully; you might get a better idea of what I am doing.
What I am presenting is a tautology (nothing is based on what you think reality is) and I think the results of that tautology are rather astounding.
Thanks for your comment -- Dick
P.S. Think of this thought problem:
You have died and gone to heaven. After playing your harp until you get bored, you ask "GOD" to explain how the universe was created. He agrees to present a lecture on the subject with one proviso: on completion of the lecture you will be given a test.
It turns out that "GOD", being the ultimate of everything is the most boring lecture concievable and you end up sleeping through the entire lecture. As you wake, he is explaining the test.
He is going to create a universe of his own design and place you in that universe to test your ability to rationally work out the rules of his construct.
He also tells you that he will do his utmost to creat a universe which will confound you (and the guy is damm good); however he will allow you to remember everything you currently know though none of it will necessarily apply to the universe you will find yourself in (lets say you will remember it as we remember dreams - as dreams seem to make sense when you are dreaming but not after you wake).
The fundamental question is: exactly what preparations can you make for this problem.
Most people would answer "None!" but I think my arguments are a substantial start.