too many facts

Ludwig Wittgenstein began to write his Tractatus almost exactly hundred years ago and the little book would become quite important for the development of positivism and analytic philosophy.
It begins with proposition 1: "Die Welt ist alles was der Fall ist."
This is usually translated as "The world is all that is the case."
followed by proposition 1.1: "The world is the totality of facts, not of things."

But I think there is a problem with this proposition.
Let us begin with a simple fact, e.g. F1 = "I am."
This is a truly basic fact and I can immediately add another one: F2 = "I know that I am, i.e. I know F1."
But there is now another fact: F3 = "There is more than 1 fact."
We can now construct an infinite number of facts from these three facts: F_n = "There are more than n-2 facts." with n=4,5,...
An interesting question is if we can count all facts and I think the answer is No.
Let us assume that all facts form a countable set {F1, F2, F3, ...}
We then have another fact for each subset; e.g. F' = "{F1, F2, F3} is a subset of all facts."
But the power set of a set S (the set of all subsets of S) always has higher cardinality than S itself.
Therefore one cannot count all facts and in fact (!) they do not even form a well-defined set.

The set of all finite English sentences is countable and so is the set of all Turing machines. In other words, neither mathematics nor the English language is capable to handle the world of all facts as envisioned by Ludwig Wittgenstein (*).
Unfortunately, he never really discusses this.

I am aware, that almost all facts stated above are statements about other facts without reference to much else. We never really got beyond "I am." and "I know that I am." and "I know that I know ... that I am."
It would seem that something like Russell's type theory would be needed here.
But the Tractatus does not do that, so I recommend to jump straight to proposition 7 and skip all the stuff in the middle.


(*) A while ago I showed that a seemingly straightforward definition of "possible worlds" leads to something that cannot be handled by set theory, i.e. math as we know it.
I think it is amazing that already all the facts that follow from two basic facts are too many to be properly handled by set theory; notice that none of the facts I considered is self-referential or otherwise ambiguous.

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