Angel Speak

Let $S$ be the set of all meaningful sentences. For the sake of argument, let us agree that a meaningful sentence is a sentence that does not defy logic and that is possible.

Statement 1. $S$ is infinite
The English sentence I have one apple implies that you can have sentences of the form I have $x$ apple(s), where $x$ is a natural number ($x \in \mathbb{N}$). Let us denote the set of such sentences claiming the possession of a number of apples as

$$A = \{\text{I have $x$ apple(s)} \mid x\in \mathbb{N} \}$$

There is no upper bound on $x$, so $|A|=\aleph_0$, and since $A \subset S$, $S$ is infinite. $\blacksquare$

Statement 2. $S$ is uncountable (i.e., $|S|\geq 2^{\aleph_0}$)
The English sentence I have a one-meter rod, since the length of a rod can be arbitrary, implies that you can have sentences of the form I have an $x$-meter-long rod, where $x$ is a real number. Let us denote the set of such sentences claiming the possession of a rod of arbitrary length (any real number) as

$$L = \{\text{I have an $x$-meter-long rod} \mid x\in \mathbb{R}\}$$

Then $|L|=2^{\aleph_0}$, and since $L\subset S$, $|S| \geq 2^{\aleph_0}$. $\blacksquare$

Examples of elements of $L$ are sentences like:

I have a 42-meter rod
I have a $\sqrt{2}$-meter rod
I have a $\pi$-meter rod
I have a googol-meter rod

[Work in progress]