How should one think of infinity.

Question

I am currently taking a first year course in mathematics and have become a bit confused when confronted with "infinity".

We are covering logic and sets and I had the following homework assignment. State if the following statement is true or false: $$\forall X\subseteq \mathbb{N},\exists n\in \mathbb{Z},\left| X \right| = n$$

I answered true, but the solution (from Hammock's Book of Proof) says its false since there can exist an X holding infinitely many elements and thus no n for which |X|=n. But, as I understood the concept, if infinity is some arbitrarily large value, then there still exists some value that represents that arbitrarily large number, i.e. would n not also be arbitrarily large in this case?

Clearly I am missing some intuition when it comes to thinking of infinity, so how should one think of it?

Answer 1

To elaborate on the comments: the set $\Bbb N$ of positive integers by definition just includes the familiar finite numbers 1, 2, 3, etc. - there are no "infinitely large values" or anything like that in $\Bbb N$. However, there are infinitely many numbers in $\Bbb N$. "Infinitely many" just means "not finitely many", i.e. there is no number $n\in\Bbb N$ such that $|\Bbb N|=n$. If there were, we would have $n=|\Bbb N|\ge|\{1,...,n+1\}|=n+1$, which is not true of any finite number $n$.

Answer 2

Infinity, just like numbers and other mathematical concepts, has different ways of looking at it. The first perspective -for infinity- a student is usually introduced to is that of "a process that does not end". To be rigorous; Take the negation of the sentence $\forall X\subseteq \mathbb{N},\exists n\in \mathbb{Z},\left| X \right| = n$, namely $\exists X\subseteq \mathbb{N},\forall n\in \mathbb{Z},\left| X \right| \neq n$. This sentence is true, since it is the negation of the one you provided, which is false. Notice what it is saying. It is saying that I can find a subset of natural numbers, such that it is impossible to count how many elements it has, i.e. the process of counting the elements never ends.