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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?

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    $\begingroup$ For the homework, take $X=\Bbb N$. Then saying that $n\in \Bbb Z$ means that $n$ is finite. So certainly $|\Bbb N|=n$ is impossible for all $n\in \Bbb Z$. $\endgroup$ May 28 at 8:17
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    $\begingroup$ The real question is why you seem to think $\infty \in \mathbb{Z}$, OP. Because ultimately, that's what your argument amounts to (irrespective of any of this handwaving about "arbitrarily large numbers"). $\endgroup$ May 28 at 8:29
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    $\begingroup$ In answer to your title, you should learn how to restate comments about infinity so as not to mention it. For example, "$X$ is an infinite set" means "for all $n\in N$, some subset of $X$ has $n$ elements". $\endgroup$
    – J.G.
    May 28 at 8:41
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    $\begingroup$ There are many infinite subsets of the natural numbers (the naturals themselves, squares, primes ... ). In fact there are uncountably many infinite subsets whilst there are countably many finite subsets (an infinite number which can be put into a bijection with the positive integers). Cantor's classical diagonal argument shows that the concept of "infinity" is not as simple as having an extra number. In some situations the symbol $\infty $ can be given a precise and useful meaning. In others it can be dangerously ambiguous and conceptually confusing. $\endgroup$ May 28 at 8:42
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    $\begingroup$ Does this answer your question? What is the definition for an infinite set? $\endgroup$ May 30 at 4:16

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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 $n$.

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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.

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