A set with the largest possible cardinality?

Question

I know that given a set $S$ the power set of $S$ has larger cardinality than $S$,

However, consider the set $T$ of all possible true statements, and let $X$ be any non-empty set. Then the function $f:X \rightarrow T$ given by the mapping for $x \in X$, $f(x)$ maps $x$ to the true statement "$x$ is an element of $X$".

doesn't $f$ give an injection from $X \rightarrow T$ implying that every set has cardinality less than or equal to $T$?

Or does the set of all possible true statements contradict one of the axioms of set theory and thus cannot be a set?

Answer 1

Ignoring for a moment issues about truth definability, or other meta-theoretical difficulties. Let's attack this heads on.

If $f$ is a function, what is its domain? Well, since $x\in X$ is a statement for every $x$, it means that the domain is everything. But you already know that "everything" does not make a set, via the usual paradoxes à la Russell, Cantor, Burali-Forti, etc.

Answer 2

Either you take as (true or not) statement something like well-formed formulae in the language of set theory. Then there are only countably many such statement, but you cannot express $x\in X$ for every set $X$ simply because in general you do not have a way to describe $X$.

Or you allow/consider only such sets $X$ that have an explicit description in the language of set theory. Then you will only have countably many such definable sets and no contradiction arises.

Or you allow arbitrary sets and extend your language with a symbol for every set. That gives you problems because already the set of symbols of your language is a proper class and in the end so is $T$.