We will never have a TOE in the Tegmarkian sense by a simple counting argument

This post is meant to argue against a Tegmarkian TOE which he calls a complete description of reality.  Towards finding a complete description of reality, one might just try a list of properties of this TOE such as its length and such as its descriptive power.  Are there zillions of concepts and experiences that we just haven't encountered so we have no words for them?  Could a zillion new words be added to the dictionary?

How about infinitely many new words?  Can reality ever be completely described if the dictionary is infinite?

Here is how I think of part of it: I believe that reality is infinite.  There are a few ways to prove that, depending on what we'd like to assume to make the argument go through.

I believe numbers exist in an abstract sense (i.e., for the sake of this argument, this is an axiom).  If you spend much time studying infinity, you at least get introduced to the smallest and next smallest transfinite numbers.  The smallest transfinite number is called Aleph Null or Aleph Zero and it is how big the set of all natural numbers is.  The next transfinite number is that of the set of real numbers (under the right side-axioms).  That means that in Hilbert's Hotel of infinite rooms, if each room had a natural number designation (or "address" (think computer memory)) AND if each room is filled, there is no way to accommodate the bus load of new patrons, each indexed by a real number.

The set of all natural numbers is said to be countably infinite.  The set of real numbers is said to be uncountable, as is every type of infinity beyond this.

The kicker is that, without going too much into formal systems, there are more real numbers than there are descriptions of all real numbers.  A description is a finite quantity but since there is no length limit inherent with descriptions, there are infinitely many different descriptions but only the countable type of infinity, that of the natural numbers.  Since there are more real numbers than that, (recall it is uncountable while the set of natural numbers is countably infinite), there are more real numbers than can ever be described.

If reality is at least as infinite as the set of real numbers, then we are done: we will never have a complete description of reality, no TOE.  It may be less obvious in the other cases: (1) If reality is finite then since you can add infinitely many neologisms to describe a finite reality, and most descriptions of reality won't even be needed, given that there are finitely many real things.  Finitely many things can be completely described.  (2) if reality is infinite, then it is possible that reality can be completely described and it is possible that reality cannot be completely described.  If the latter, we are done again since that would imply there is no TOE.  The trickiest case as I see it is what happens with when reality is infinite and can be completely described. Then a TOE would exist, although it may take eons to understand to even write it down, like Graham's number.

So I'm basically saying that a simple counting argument torpedos the notion of a usable TOE defined as a complete description of reality.  Sad to say it but that's where the investigation goes.  I still maintain hope that a TOE can not only exist but translated into more normal speech.