Can't understand the meaning of this simple statement

Question

I am trying to study Logic and I reached quantifiers. I understand how to translate this statement and can read it, but I can't wrap my head around what it means.

$\exists n \in \mathbb N, \forall \mathit X \in \mathscr P(\mathbb N), |\mathit X| < n$

Answer 1

One way to read these statements is to work from the inside out.

Let's start with the innermost part: "$|X|<n$". This says

The set $X$ has less than $n$ elements.

Good so far.

Next, let's add in the universal quantifier: "$\forall X \in \mathcal P(\mathbb N), |X| < n$". This says:

Every subset of the natural numbers has less than $n$ elements.

Now my suspicions start to rise, so before going further I'm going to sit and think about this statement, just because I want to (and because it enchances my understanding).

I know that this statement is false if I substitute $n=100$, because I can produce a counterexample, i.e. I can produce a subset $X \subset \mathbb N$ with exactly $100$ elements, namely $X = \{1,2,3,4,...,100\}$. In fact, I can immediately see that this statement is false no matter which natural number I substitute for $n$, because I can produce a subset $X \subset \mathbb N$ with exactly $n$ elements, namely $\{1,2,3,4,...,n\}$. Still, though, if I hadn't peeked at that existential quantifier, I might have had the thought "but the statement is true if I substitute an infinite cardinal number in place of $n$".

However, now I look at the whole statement: "$\exists n \in \mathbb N, \forall \mathit X \in \mathscr P(\mathbb N), |\mathit X| < n$". This says:

There is a natural number $n$ such that every subset of $\mathbb N$ has less than $n$ elements.

So that's the answer to your question. Furthermore, I know that statement to be false, as just explained above.

Answer 2

The statement reads, "There exists a natural number $n$ such that for all $X$ in the power set of $\mathbb{N}$ (the set of all natural numbers), the cardinality of $X$ is strictly less than $n$." This means that if you choose some natural number $n$, you can always find a subset of the power set of $\mathbb{N}$ - which is basically a subset of $\mathbb{N}$ - such that it has fewer than $n$ elements.